<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
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      justify-content: center;
      min-height: 100%;
      padding: 12px 16px;
      user-select: none;
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      cursor: pointer;
      transition: background 0.2s, border-color 0.2s, color 0.2s;
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      background: #e2e8f0;
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      background: #1e2945;
      color: #ffffff;
      border-color: #1e2945;
    }

    .svg-container {
      width: 100%;
      max-width: 580px;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    .domain-row {
      transition: opacity 0.35s ease;
    }
    .dimmed {
      opacity: 0.2;
    }
    .highlighted {
      opacity: 1;
    }

    .guide-line {
      stroke: #cbd5e1;
      stroke-dasharray: 4 4;
      stroke-width: 1.5;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="btn active" id="btn-all" onclick="setMode('all')">Show Both & Intersection</button>
    <button class="btn" id="btn-f" onclick="setMode('f')">f(x) Domain</button>
    <button class="btn" id="btn-g" onclick="setMode('g')">g(x) Domain</button>
    <button class="btn" id="btn-inter" onclick="setMode('inter')">Intersection [0, 6]</button>
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  <div class="svg-container">
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      <line id="guide-6" class="guide-line" x1="410" y1="35" x2="410" y2="280" />

      
      <g id="row-f" class="domain-row">
        <text x="50" y="42" font-size="14" font-weight="700" fill="#1e2945">f(x) = √x</text>
        <text x="50" y="60" font-size="13" font-weight="600" fill="#0284c7">[0, ∞)</text>
        
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      </g>

      
      <g id="row-g" class="domain-row">
        <text x="50" y="118" font-size="14" font-weight="700" fill="#1e2945">g(x) = √(6 − x)</text>
        <text x="50" y="136" font-size="13" font-weight="600" fill="#d97706">(−∞, 6]</text>
        
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      </g>

      
      <g id="row-inter" class="domain-row">
        <text x="50" y="196" font-size="14" font-weight="700" fill="#1e2945">f ∩ g</text>
        <text x="50" y="214" font-size="13" font-weight="700" fill="#16a34a">[0, 6]</text>
        
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      </g>
    </svg>
  </div>

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      const rowF = document.getElementById('row-f');
      const rowG = document.getElementById('row-g');
      const rowInter = document.getElementById('row-inter');

      if (mode === 'all') {
        document.getElementById('btn-all').classList.add('active');
        rowF.className = 'domain-row highlighted';
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        rowInter.className = 'domain-row highlighted';
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        document.getElementById('btn-f').classList.add('active');
        rowF.className = 'domain-row highlighted';
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        document.getElementById('btn-inter').classList.add('active');
        rowF.className = 'domain-row dimmed';
        rowG.className = 'domain-row dimmed';
        rowInter.className = 'domain-row highlighted';
      }
    }
  </script>
</body>
</html>

Arithmetic Combinations of Functions and Their Domains

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px 14px;
      user-select: none;
    }
    .top-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      flex-wrap: wrap;
      width: 100%;
      margin-bottom: 8px;
    }
    .fn-badge {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 4px 10px;
      font-size: 14px;
      font-weight: 500;
      color: #334155;
    }
    .tabs {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 480px;
      justify-content: center;
      margin-bottom: 10px;
    }
    .tab-btn {
      flex: 1;
      padding: 8px 4px;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      background: #f8fafc;
      color: #475569;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s, box-shadow 0.2s, color 0.2s, border-color 0.2s;
    }
    .tab-btn:hover {
      background: #e2e8f0;
    }
    .tab-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }
    .detail-card {
      width: 100%;
      max-width: 480px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 12px 16px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      min-height: 105px;
      justify-content: center;
    }
    .row-math {
      display: flex;
      align-items: center;
      justify-content: space-between;
      flex-wrap: wrap;
      gap: 6px;
    }
    .formula-box {
      font-size: 16px;
      font-weight: 600;
      color: #1e2945;
    }
    .domain-badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      background: #ecfdf5;
      border: 1px solid #a7f3d0;
      color: #065f46;
      font-size: 13px;
      font-weight: 600;
      padding: 3px 8px;
      border-radius: 6px;
    }
    .restriction-box {
      font-size: 13px;
      color: #b91c1c;
      background: #fef2f2;
      border: 1px solid #fecaca;
      border-radius: 6px;
      padding: 4px 8px;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .number-line-container {
      width: 100%;
      max-width: 500px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 12px 6px 12px;
      margin-top: 8px;
    }
    .nl-header {
      font-size: 12px;
      font-weight: 700;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-bottom: 2px;
      text-align: center;
    }
    svg {
      width: 100%;
      height: 70px;
      display: block;
      overflow: visible;
    }
  </style>
</head>
<body>
  <div class="top-bar">
    <div class="fn-badge" id="fnF"></div>
    <div class="fn-badge" id="fnG"></div>
  </div>

  <div class="tabs">
    <button class="tab-btn active" id="btn0" onclick="setOp(0)">(f + g)(x)</button>
    <button class="tab-btn" id="btn1" onclick="setOp(1)">(f · g)(x)</button>
    <button class="tab-btn" id="btn2" onclick="setOp(2)">(f / g)(x)</button>
  </div>

  <div class="detail-card">
    <div class="row-math">
      <div class="formula-box" id="formulaDisplay"></div>
      <div class="domain-badge" id="domainDisplay"></div>
    </div>
    <div class="restriction-box" id="restrictionDisplay" style="display: none;"></div>
  </div>

  <div class="number-line-container">
    <div class="nl-header" id="nlTitle">Domain on Real Line</div>
    <svg viewBox="0 0 440 64" id="nlSvg"></svg>
  </div>

  <script>
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      {
        formulaTex: "(f + g)(x) = \\sqrt{x + 3} + x - 2",
        domainTex: "[-3, \\infty)",
        restriction: null,
        hasHole: false
      },
      {
        formulaTex: "(f \\cdot g)(x) = (x - 2)\\sqrt{x + 3}",
        domainTex: "[-3, \\infty)",
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      },
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        formulaTex: "\\left(\\frac{f}{g}\\right)(x) = \\frac{\\sqrt{x + 3}}{x - 2}",
        domainTex: "[-3, 2) \\cup (2, \\infty)",
        restriction: "x - 2 \\neq 0 \\implies x \\neq 2",
        hasHole: true
      }
    ];

    function renderMath(element, tex) {
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        katex.render(tex, element, { throwOnError: false });
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      }
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    renderMath(document.getElementById('fnF'), "f(x) = \\sqrt{x + 3}");
    renderMath(document.getElementById('fnG'), "g(x) = x - 2");

    let currentOp = 0;

    function renderNumberLine(hasHole) {
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      const axisY = 32;

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      const x2 = toSvgX(2);
      const xEnd = toSvgX(maxX) + 12;

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        intervalSvg = `
          
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          <polygon points="${xEnd + 4},${axisY} ${xEnd - 4},${axisY - 4} ${xEnd - 4},${axisY + 4}" fill="#2563eb" />
          
          <circle cx="${xMinus3}" cy="${axisY}" r="5" fill="#2563eb" stroke="#ffffff" stroke-width="1.5" />
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      } else {
        intervalSvg = `
          
          <line x1="${xMinus3}" y1="${axisY}" x2="${x2 - 5}" y2="${axisY}" stroke="#2563eb" stroke-width="5" stroke-linecap="round" />
          
          <line x1="${x2 + 5}" y1="${axisY}" x2="${xEnd}" y2="${axisY}" stroke="#2563eb" stroke-width="5" stroke-linecap="round" />
          
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          <circle cx="${xMinus3}" cy="${axisY}" r="5" fill="#2563eb" stroke="#ffffff" stroke-width="1.5" />
          
          <circle cx="${x2}" cy="${axisY}" r="5" fill="#ffffff" stroke="#ef4444" stroke-width="2.5" />
          
          <text x="${x2}" y="${axisY - 10}" text-anchor="middle" font-size="11" font-weight="700" fill="#ef4444">x ≠ 2</text>
        `;
      }

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            <path d="M 0 0 L 6 3 L 0 6 z" fill="#cbd5e1" />
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        </defs>
        
        <line x1="${originX - 12}" y1="${axisY}" x2="${originX + axisLen + 15}" y2="${axisY}" stroke="#cbd5e1" stroke-width="2" marker-end="url(#axisArrow)" />
        ${ticksSvg}
        ${intervalSvg}
      `;

      document.getElementById('nlSvg').innerHTML = svgContent;
    }

    function setOp(index) {
      currentOp = index;
      for (let i = 0; i < 3; i++) {
        const btn = document.getElementById('btn' + i);
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          btn.classList.add('active');
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          btn.classList.remove('active');
        }
      }

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      renderMath(document.getElementById('formulaDisplay'), item.formulaTex);
      
      const domainEl = document.getElementById('domainDisplay');
      domainEl.innerHTML = '<span style="font-size:11px; color:#047857; text-transform:uppercase; font-weight:700;">Domain:</span> ';
      const span = document.createElement('span');
      renderMath(span, item.domainTex);
      domainEl.appendChild(span);

      const restrEl = document.getElementById('restrictionDisplay');
      if (item.restriction) {
        restrEl.style.display = 'flex';
        restrEl.innerHTML = '<span style="font-weight:700;">Restriction:</span> ';
        const rSpan = document.createElement('span');
        renderMath(rSpan, item.restriction);
        restrEl.appendChild(rSpan);
      } else {
        restrEl.style.display = 'none';
      }

      renderNumberLine(item.hasHole);
    }

    // Initialize
    setOp(0);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      background: transparent;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      padding: 10px;
      display: flex;
      justify-content: center;
      align-items: center;
      min-height: 100%;
    }
    .grid-container {
      display: grid;
      grid-template-columns: repeat(2, 1fr);
      gap: 14px;
      width: 100%;
      max-width: 620px;
    }
    @media (max-width: 440px) {
      .grid-container {
        grid-template-columns: 1fr;
        gap: 10px;
      }
    }
    .panel {
      background: #f8fafc;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      padding: 14px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      position: relative;
      min-height: 125px;
      border-top-width: 4px;
    }
    .panel-sum {
      border-top-color: #4f46e5;
    }
    .panel-diff {
      border-top-color: #0d9488;
    }
    .panel-prod {
      border-top-color: #d97706;
    }
    .panel-quot {
      border-top-color: #e11d48;
    }
    .tag {
      font-size: 11px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      padding: 3px 8px;
      border-radius: 6px;
      margin-bottom: 8px;
    }
    .tag-sum {
      background: #e0e7ff;
      color: #4338ca;
    }
    .tag-diff {
      background: #ccfbf1;
      color: #0f766e;
    }
    .tag-prod {
      background: #fef3c7;
      color: #b45309;
    }
    .tag-quot {
      background: #ffe4e6;
      color: #be123c;
    }
    .formula {
      font-size: 16px;
      margin: 4px 0;
      text-align: center;
    }
    .helper {
      font-size: 12px;
      color: #64748b;
      font-weight: 500;
      margin-top: 6px;
      text-align: center;
    }
    .restriction-badge {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      background: #ffe4e6;
      border: 1px solid #fecdd3;
      color: #e11d48;
      font-size: 11px;
      font-weight: 700;
      padding: 2px 7px;
      border-radius: 9999px;
      margin-top: 6px;
    }
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</head>
<body>

<div class="grid-container">
  
  <div class="panel panel-sum">
    <span class="tag tag-sum">+ Sum</span>
    <div class="formula" id="math-sum"></div>
    <div class="helper">Add outputs</div>
  </div>

  
  <div class="panel panel-diff">
    <span class="tag tag-diff">− Difference</span>
    <div class="formula" id="math-diff"></div>
    <div class="helper">Subtract outputs</div>
  </div>

  
  <div class="panel panel-prod">
    <span class="tag tag-prod">× Product</span>
    <div class="formula" id="math-prod"></div>
    <div class="helper">Multiply outputs</div>
  </div>

  
  <div class="panel panel-quot">
    <span class="tag tag-quot">÷ Quotient</span>
    <div class="formula" id="math-quot"></div>
    <div class="restriction-badge">
      <span id="math-restr"></span>
    </div>
    <div class="helper">Divide outputs</div>
  </div>
</div>

<script>
  window.addEventListener('DOMContentLoaded', () => {
    if (typeof katex !== 'undefined') {
      katex.render("(f + g)(x) = f(x) + g(x)", document.getElementById('math-sum'), { throwOnError: false });
      katex.render("(f - g)(x) = f(x) - g(x)", document.getElementById('math-diff'), { throwOnError: false });
      katex.render("(f \\cdot g)(x) = f(x) \\cdot g(x)", document.getElementById('math-prod'), { throwOnError: false });
      katex.render("\\left(\\frac{f}{g}\\right)(x) = \\frac{f(x)}{g(x)}", document.getElementById('math-quot'), { throwOnError: false });
      katex.render("g(x) \\neq 0", document.getElementById('math-restr'), { throwOnError: false });
    }
  });
</script>

</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }
    .header-control {
      display: flex;
      align-items: center;
      gap: 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      padding: 6px 14px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
    }
    .header-control input[type=range] {
      width: 130px;
      accent-color: #2563eb;
      cursor: pointer;
    }
    .graphs-container {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 12px;
      width: 100%;
      max-width: 620px;
    }
    .graph-card {
      flex: 1 1 260px;
      max-width: 295px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }
    .graph-title {
      font-size: 15px;
      font-weight: 700;
      margin-bottom: 4px;
    }
    svg {
      width: 100%;
      height: 190px;
      display: block;
      overflow: visible;
    }
    .badge {
      margin-top: 8px;
      padding: 4px 10px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 700;
      display: flex;
      align-items: center;
      gap: 4px;
      text-align: center;
    }
    .badge-fail {
      background: #fef2f2;
      color: #dc2626;
      border: 1px solid #fecaca;
    }
    .badge-pass {
      background: #f0fdf4;
      color: #16a34a;
      border: 1px solid #bbf7d0;
    }
    .hint {
      font-size: 11px;
      color: #64748b;
      margin-top: 2px;
    }
  </style>
</head>
<body>

  <div class="header-control">
    <span>Horizontal Line: <span id="yValText" style="color: #2563eb;">y = 3.0</span></span>
    <input type="range" id="ySlider" min="-3" max="4.5" step="0.1" value="3">
  </div>

  <div class="graphs-container">
    
    <div class="graph-card">
      <div class="graph-title">f(x) = x²</div>
      <svg id="svg1" viewBox="0 0 200 180">
        
        <g stroke="#f1f5f9" stroke-width="1">
          <line x1="20" y1="20" x2="20" y2="170" />
          <line x1="60" y1="20" x2="60" y2="170" />
          <line x1="100" y1="20" x2="100" y2="170" />
          <line x1="140" y1="20" x2="140" y2="170" />
          <line x1="180" y1="20" x2="180" y2="170" />
          <line x1="10" y1="35" x2="190" y2="35" />
          <line x1="10" y1="65" x2="190" y2="65" />
          <line x1="10" y1="95" x2="190" y2="95" />
          <line x1="10" y1="125" x2="190" y2="125" />
          <line x1="10" y1="155" x2="190" y2="155" />
        </g>
        
        <line x1="10" y1="155" x2="190" y2="155" stroke="#94a3b8" stroke-width="1.2" />
        <line x1="100" y1="10" x2="100" y2="175" stroke="#94a3b8" stroke-width="1.2" />
        
        <text x="188" y="167" font-size="10" fill="#94a3b8" font-family="sans-serif">x</text>
        <text x="105" y="18" font-size="10" fill="#94a3b8" font-family="sans-serif">y</text>
        
        
        <path id="curve1" stroke="#2563eb" stroke-width="2.5" fill="none" stroke-linecap="round" />

        
        <line id="line1" stroke="#64748b" stroke-width="1.8" stroke-dasharray="4,3" />

        
        <g id="dots1"></g>
      </svg>
      <div id="badge1" class="badge badge-fail">✕ 2 Points → Not Invertible</div>
      <div class="hint" id="hint1">Fails Horizontal Line Test</div>
    </div>

    
    <div class="graph-card">
      <div class="graph-title">g(x) = x³</div>
      <svg id="svg2" viewBox="0 0 200 180">
        
        <g stroke="#f1f5f9" stroke-width="1">
          <line x1="30" y1="10" x2="30" y2="170" />
          <line x1="65" y1="10" x2="65" y2="170" />
          <line x1="100" y1="10" x2="100" y2="170" />
          <line x1="135" y1="10" x2="135" y2="170" />
          <line x1="170" y1="10" x2="170" y2="170" />
          <line x1="10" y1="30" x2="190" y2="30" />
          <line x1="10" y1="60" x2="190" y2="60" />
          <line x1="10" y1="90" x2="190" y2="90" />
          <line x1="10" y1="120" x2="190" y2="120" />
          <line x1="10" y1="150" x2="190" y2="150" />
        </g>
        
        <line x1="10" y1="90" x2="190" y2="90" stroke="#94a3b8" stroke-width="1.2" />
        <line x1="100" y1="10" x2="100" y2="170" stroke="#94a3b8" stroke-width="1.2" />
        
        <text x="188" y="102" font-size="10" fill="#94a3b8" font-family="sans-serif">x</text>
        <text x="105" y="18" font-size="10" fill="#94a3b8" font-family="sans-serif">y</text>

        
        <path id="curve2" stroke="#2563eb" stroke-width="2.5" fill="none" stroke-linecap="round" />

        
        <line id="line2" stroke="#64748b" stroke-width="1.8" stroke-dasharray="4,3" />

        
        <g id="dots2"></g>
      </svg>
      <div id="badge2" class="badge badge-pass">✓ 1 Point → Invertible</div>
      <div class="hint">Passes Horizontal Line Test</div>
    </div>
  </div>

  <script>
    // Scale mappings
    // Graph 1: x in [-2.5, 2.5] -> [0, 200], y in [-0.5, 5] -> [170, 5]
    // origin1 = (100, 155), scaleX1 = 38, scaleY1 = 30
    function toSvg1(x, y) {
      return { cx: 100 + x * 38, cy: 155 - y * 30 };
    }

    // Graph 2: x in [-1.8, 1.8] -> [0, 200], y in [-4.5, 4.5] -> [170, 10]
    // origin2 = (100, 90), scaleX2 = 45, scaleY2 = 17
    function toSvg2(x, y) {
      return { cx: 100 + x * 45, cy: 90 - y * 17 };
    }

    // Render static curve 1: y = x^2
    let d1 = "";
    for (let x = -2.15; x <= 2.15; x += 0.05) {
      const y = x * x;
      const pt = toSvg1(x, y);
      d1 += (d1 === "" ? "M " : " L ") + pt.cx.toFixed(1) + " " + pt.cy.toFixed(1);
    }
    document.getElementById("curve1").setAttribute("d", d1);

    // Render static curve 2: y = x^3
    let d2 = "";
    for (let x = -1.65; x <= 1.65; x += 0.04) {
      const y = Math.pow(x, 3);
      const pt = toSvg2(x, y);
      d2 += (d2 === "" ? "M " : " L ") + pt.cx.toFixed(1) + " " + pt.cy.toFixed(1);
    }
    document.getElementById("curve2").setAttribute("d", d2);

    const slider = document.getElementById("ySlider");
    const yValText = document.getElementById("yValText");
    const line1 = document.getElementById("line1");
    const line2 = document.getElementById("line2");
    const dots1 = document.getElementById("dots1");
    const dots2 = document.getElementById("dots2");
    const badge1 = document.getElementById("badge1");
    const hint1 = document.getElementById("hint1");

    function update() {
      const y = parseFloat(slider.value);
      yValText.textContent = "y = " + y.toFixed(1);

      // Line 1
      const p1 = toSvg1(0, y);
      line1.setAttribute("x1", "10");
      line1.setAttribute("y1", p1.cy);
      line1.setAttribute("x2", "190");
      line1.setAttribute("y2", p1.cy);

      // Intersections for y = x^2
      dots1.innerHTML = "";
      if (y > 0.05) {
        const xPos = Math.sqrt(y);
        const ptA = toSvg1(-xPos, y);
        const ptB = toSvg1(xPos, y);
        dots1.innerHTML = `
          <circle cx="${ptA.cx}" cy="${ptA.cy}" r="5.5" fill="#ef4444" stroke="#ffffff" stroke-width="1.5" />
          <circle cx="${ptB.cx}" cy="${ptB.cy}" r="5.5" fill="#ef4444" stroke="#ffffff" stroke-width="1.5" />
        `;
        badge1.className = "badge badge-fail";
        badge1.textContent = "✕ 2 Points → Not Invertible";
        hint1.textContent = "Multiple inputs share output y = " + y.toFixed(1);
      } else if (Math.abs(y) <= 0.05) {
        const pt0 = toSvg1(0, 0);
        dots1.innerHTML = `<circle cx="${pt0.cx}" cy="${pt0.cy}" r="5.5" fill="#f59e0b" stroke="#ffffff" stroke-width="1.5" />`;
        badge1.className = "badge badge-fail";
        badge1.textContent = "✕ Fails Test Elsewhere";
        hint1.textContent = "Must pass for every horizontal line";
      } else {
        badge1.className = "badge badge-fail";
        badge1.textContent = "✕ 0 Points at y < 0";
        hint1.textContent = "Fails test when y > 0";
      }

      // Line 2
      const p2 = toSvg2(0, y);
      line2.setAttribute("x1", "10");
      line2.setAttribute("y1", p2.cy);
      line2.setAttribute("x2", "190");
      line2.setAttribute("y2", p2.cy);

      // Intersection for y = x^3 (always exactly 1)
      const xCube = Math.cbrt(y);
      const ptCube = toSvg2(xCube, y);
      dots2.innerHTML = `<circle cx="${ptCube.cx}" cy="${ptCube.cy}" r="5.5" fill="#22c55e" stroke="#ffffff" stroke-width="1.5" />`;
    }

    slider.addEventListener("input", update);
    update();
  </script>
</body>
</html>

Determining Whether a Function Is Invertible from Its Graph

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      user-select: none;
      -webkit-user-select: none;
    }

    .header-tag {
      font-size: 13px;
      font-weight: 700;
      color: #4b5563;
      margin-bottom: 6px;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .header-tag span {
      background: #eef2ff;
      color: #3b82f6;
      padding: 2px 8px;
      border-radius: 6px;
      font-family: ui-monospace, monospace;
      font-size: 12px;
    }

    .graph-container {
      position: relative;
      width: 100%;
      max-width: 440px;
      aspect-ratio: 1 / 0.85;
      background: #fafbfc;
      border: 1px solid #e5e7eb;
      border-radius: 12px;
      overflow: hidden;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    /* Slider / Control row */
    .controls {
      width: 100%;
      max-width: 440px;
      margin-top: 10px;
      display: flex;
      align-items: center;
      gap: 10px;
      padding: 6px 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
    }
    .controls label {
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      white-space: nowrap;
    }
    .controls input[type="range"] {
      flex: 1;
      accent-color: #ef4444;
      cursor: pointer;
    }

    /* Result Callout */
    .callout {
      width: 100%;
      max-width: 440px;
      margin-top: 8px;
      background: #fef2f2;
      border: 1px solid #fecaca;
      border-radius: 8px;
      padding: 8px 12px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      font-size: 13px;
      color: #991b1b;
      font-weight: 600;
    }

    .badge-no {
      background: #dc2626;
      color: #ffffff;
      padding: 3px 8px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 800;
      letter-spacing: 0.5px;
      white-space: nowrap;
    }

    .badge-yes {
      background: #16a34a;
      color: #ffffff;
      padding: 3px 8px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 800;
      letter-spacing: 0.5px;
      white-space: nowrap;
    }
  </style>
</head>
<body>

  <div class="header-tag">
    Horizontal Line Test <span>g(x) = x² − 4x + 3</span>
  </div>

  <div class="graph-container">
    <svg id="plot" viewBox="0 0 440 370">
      <defs>
        <pattern id="grid" width="40" height="40" patternUnits="userSpaceOnUse">
          <path d="M 40 0 L 0 0 0 40" fill="none" stroke="#f1f5f9" stroke-width="1.2" />
        </pattern>
        
        <marker id="arr" viewBox="0 0 6 6" refX="5" refY="3" markerWidth="4" markerHeight="4" orient="auto">
          <path d="M 0 0 L 6 3 L 0 6 z" fill="#94a3b8" />
        </marker>
      </defs>

      
      <rect x="20" y="15" width="400" height="340" fill="url(#grid)" />

      
      
      <line x1="100" y1="360" x2="100" y2="15" stroke="#94a3b8" stroke-width="1.5" marker-end="url(#arr)" />
      
      <line x1="20" y1="255" x2="420" y2="255" stroke="#94a3b8" stroke-width="1.5" marker-end="url(#arr)" />

      
      <text x="424" y="259" font-size="12" fill="#64748b" font-weight="600">x</text>
      <text x="100" y="12" font-size="12" fill="#64748b" font-weight="600" text-anchor="middle">y</text>

      
      
      <text x="45" y="272" font-size="11" fill="#94a3b8" text-anchor="middle">-1</text>
      <text x="155" y="272" font-size="11" fill="#94a3b8" text-anchor="middle">1</text>
      <text x="210" y="272" font-size="11" fill="#94a3b8" text-anchor="middle">2</text>
      <text x="265" y="272" font-size="11" fill="#94a3b8" text-anchor="middle">3</text>
      <text x="320" y="272" font-size="11" fill="#94a3b8" text-anchor="middle">4</text>

      
      <text x="88" y="313" font-size="11" fill="#94a3b8" text-anchor="end">-1</text>
      <text x="88" y="203" font-size="11" fill="#94a3b8" text-anchor="end">1</text>
      <text x="88" y="148" font-size="11" fill="#94a3b8" text-anchor="end">2</text>
      <text x="88" y="93" font-size="11" fill="#94a3b8" text-anchor="end">3</text>
      <text x="88" y="38" font-size="11" fill="#94a3b8" text-anchor="end">4</text>

      
      <path id="curve" fill="none" stroke="#2563eb" stroke-width="3" stroke-linecap="round" />

      
      <circle cx="210" cy="310" r="4.5" fill="#2563eb" />
      <text x="210" y="330" font-size="11" fill="#1e40af" font-weight="600" text-anchor="middle">Vertex (2, -1)</text>

      
      <g id="test-line-group">
        <line id="test-line" x1="20" y1="255" x2="420" y2="255" stroke="#ef4444" stroke-width="2" stroke-dasharray="6,5" />
        <text id="test-line-label" x="415" y="248" font-size="11" fill="#dc2626" font-weight="700" text-anchor="end">y = 0</text>
      </g>

      
      <g id="intersection-group"></g>
    </svg>
  </div>

  <div class="controls">
    <label for="ySlider">Test line y:</label>
    <input type="range" id="ySlider" min="-1.5" max="4.5" step="0.5" value="0">
  </div>

  <div class="callout" id="callout">
    <div id="callout-text">Two inputs (x = 1, x = 3) produce output y = 0</div>
    <div class="badge-no" id="badge">INVERTIBLE: NO</div>
  </div>

  <script>
    // Coordinate mapping:
    // x range: [-2, 5.5], x=0 is at px=100, scaleX = 55 px/unit
    // y range: [-2, 5], y=0 is at py=255, scaleY = 55 px/unit
    const xToPx = x => 100 + x * 55;
    const yToPx = y => 255 - y * 55;
    const pxToX = px => (px - 100) / 55;
    const pyToY = py => (255 - py) / 55;

    // Draw Parabola curve g(x) = x^2 - 4x + 3
    const curvePath = document.getElementById('curve');
    let d = '';
    for (let x = -0.6; x <= 4.6; x += 0.05) {
      const y = x * x - 4 * x + 3;
      const px = xToPx(x);
      const py = yToPx(y);
      if (d === '') d += `M ${px.toFixed(1)} ${py.toFixed(1)}`;
      else d += ` L ${px.toFixed(1)} ${py.toFixed(1)}`;
    }
    curvePath.setAttribute('d', d);

    const slider = document.getElementById('ySlider');
    const testLine = document.getElementById('test-line');
    const testLabel = document.getElementById('test-line-label');
    const intGroup = document.getElementById('intersection-group');
    const calloutText = document.getElementById('callout-text');
    const badge = document.getElementById('badge');

    function update(yVal) {
      const py = yToPx(yVal);
      testLine.setAttribute('y1', py);
      testLine.setAttribute('y2', py);
      testLabel.setAttribute('y', py - 7);
      testLabel.textContent = `y = ${yVal}`;

      intGroup.innerHTML = '';

      // g(x) = x^2 - 4x + 3 = yVal  => x^2 - 4x + (3 - yVal) = 0
      // Discriminant: D = 16 - 4*(3 - yVal) = 4 + 4*yVal = 4*(1 + yVal)
      const D = 4 * (1 + yVal);

      if (D > 0.001) {
        const x1 = (4 - Math.sqrt(D)) / 2;
        const x2 = (4 + Math.sqrt(D)) / 2;

        [x1, x2].forEach(x => {
          const px = xToPx(x);

          // Dot
          const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
          circle.setAttribute('cx', px);
          circle.setAttribute('cy', py);
          circle.setAttribute('r', '5.5');
          circle.setAttribute('fill', '#ef4444');
          circle.setAttribute('stroke', '#ffffff');
          circle.setAttribute('stroke-width', '1.5');
          intGroup.appendChild(circle);

          // Coordinate label
          const txt = document.createElementNS('http://www.w3.org/2000/svg', 'text');
          txt.setAttribute('x', px);
          txt.setAttribute('y', py - 10);
          txt.setAttribute('font-size', '11');
          txt.setAttribute('font-weight', '700');
          txt.setAttribute('fill', '#dc2626');
          txt.setAttribute('text-anchor', 'middle');
          txt.textContent = `(${x % 1 === 0 ? x : x.toFixed(1)}, ${yVal})`;
          intGroup.appendChild(txt);
        });

        const x1Str = (x1 % 1 === 0 ? x1 : x1.toFixed(1));
        const x2Str = (x2 % 1 === 0 ? x2 : x2.toFixed(1));
        calloutText.textContent = `Two inputs (x = ${x1Str}, x = ${x2Str}) produce output y = ${yVal}`;
        badge.className = 'badge-no';
        badge.textContent = 'INVERTIBLE: NO';
      } else if (Math.abs(D) <= 0.001) {
        const x = 2;
        const px = xToPx(x);

        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', px);
        circle.setAttribute('cy', py);
        circle.setAttribute('r', '5.5');
        circle.setAttribute('fill', '#ef4444');
        circle.setAttribute('stroke', '#ffffff');
        circle.setAttribute('stroke-width', '1.5');
        intGroup.appendChild(circle);

        calloutText.textContent = `1 input (x = 2) at vertex, but other y-values have 2`;
        badge.className = 'badge-no';
        badge.textContent = 'INVERTIBLE: NO';
      } else {
        calloutText.textContent = `No inputs produce output y = ${yVal}`;
        badge.className = 'badge-no';
        badge.textContent = 'INVERTIBLE: NO';
      }
    }

    slider.addEventListener('input', e => update(parseFloat(e.target.value)));
    update(0);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .toolbar {
      display: flex;
      gap: 8px;
      margin-bottom: 12px;
      background: #f1f5f9;
      padding: 4px;
      border-radius: 20px;
    }

    .toggle-btn {
      border: none;
      background: transparent;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      border-radius: 16px;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .toggle-btn.active {
      background: #ffffff;
      color: #1e2945;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.08);
    }

    .grid-container {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      max-width: 580px;
    }

    @media (max-width: 440px) {
      .grid-container {
        grid-template-columns: 1fr;
        gap: 10px;
      }
    }

    .panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 14px;
      padding: 12px;
      box-shadow: 0 4px 12px rgba(30, 41, 69, 0.04);
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    .panel-header {
      font-size: 13px;
      font-weight: 700;
      text-align: center;
      margin-bottom: 10px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .one-to-one .panel-header {
      color: #059669;
    }

    .two-to-one .panel-header {
      color: #dc2626;
    }

    .map-stage {
      position: relative;
      width: 100%;
      height: 150px;
    }

    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    .col-label {
      font-size: 11px;
      font-weight: 700;
      fill: #64748b;
      text-anchor: middle;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    .node-bg {
      fill: #f8fafc;
      stroke: #cbd5e1;
      stroke-width: 1.5;
      transition: all 0.3s ease;
    }

    .node-val {
      font-size: 13px;
      font-weight: 700;
      fill: #1e2945;
      text-anchor: middle;
      dominant-baseline: central;
    }

    .map-arrow {
      stroke-width: 2;
      fill: none;
      transition: stroke 0.3s ease;
    }

    .arrow-green {
      stroke: #10b981;
    }

    .arrow-red {
      stroke: #ef4444;
    }

    .warning-cue {
      opacity: 0;
      transition: opacity 0.3s ease;
    }

    body.reverse-mode .warning-cue {
      opacity: 1;
    }

    .badge-status {
      font-size: 11px;
      font-weight: 600;
      padding: 3px 8px;
      border-radius: 6px;
      margin-top: 6px;
    }

    .badge-pass {
      background: #ecfdf5;
      color: #047857;
      border: 1px solid #a7f3d0;
    }

    .badge-fail {
      background: #fef2f2;
      color: #b91c1c;
      border: 1px solid #fecaca;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="toggle-btn active" id="btnFwd" onclick="setMode('forward')">Forward: f(x)</button>
    <button class="toggle-btn" id="btnRev" onclick="setMode('reverse')">Reverse: f⁻¹(x)</button>
  </div>

  <div class="grid-container">
    
    <div class="panel one-to-one">
      <div class="panel-header">
        <span>One-to-One</span>
      </div>

      <div class="map-stage">
        <svg viewBox="0 0 200 150">
          <defs>
            
            <marker id="arr-g" markerWidth="6" markerHeight="6" refX="5" refY="3" orient="auto">
              <path d="M0,0 L6,3 L0,6 Z" fill="#10b981" />
            </marker>
            
            <marker id="arr-g-rev" markerWidth="6" markerHeight="6" refX="1" refY="3" orient="auto">
              <path d="M6,0 L0,3 L6,6 Z" fill="#10b981" />
            </marker>
          </defs>

          
          <text x="45" y="16" class="col-label" id="lbl-left-d">Domain</text>
          <text x="155" y="16" class="col-label" id="lbl-left-r">Range</text>

          
          <path id="arrow-1" d="M 62 42 L 138 42" class="map-arrow arrow-green" marker-end="url(#arr-g)" />
          <path id="arrow-2" d="M 62 82 L 138 82" class="map-arrow arrow-green" marker-end="url(#arr-g)" />
          <path id="arrow-3" d="M 62 122 L 138 122" class="map-arrow arrow-green" marker-end="url(#arr-g)" />

          
          <g transform="translate(45, 42)">
            <circle class="node-bg" r="16" />
            <text class="node-val">1</text>
          </g>
          <g transform="translate(45, 82)">
            <circle class="node-bg" r="16" />
            <text class="node-val">2</text>
          </g>
          <g transform="translate(45, 122)">
            <circle class="node-bg" r="16" />
            <text class="node-val">3</text>
          </g>

          
          <g transform="translate(155, 42)">
            <circle class="node-bg" r="16" />
            <text class="node-val">2</text>
          </g>
          <g transform="translate(155, 82)">
            <circle class="node-bg" r="16" />
            <text class="node-val">4</text>
          </g>
          <g transform="translate(155, 122)">
            <circle class="node-bg" r="16" />
            <text class="node-val">6</text>
          </g>
        </svg>
      </div>

      <div class="badge-status badge-pass" id="badge-left">✓ Reversible</div>
    </div>

    
    <div class="panel two-to-one">
      <div class="panel-header">
        <span>Two-to-One</span>
      </div>

      <div class="map-stage">
        <svg viewBox="0 0 200 150">
          <defs>
            
            <marker id="arr-r" markerWidth="6" markerHeight="6" refX="5" refY="3" orient="auto">
              <path d="M0,0 L6,3 L0,6 Z" fill="#ef4444" />
            </marker>
            
            <marker id="arr-r-rev" markerWidth="6" markerHeight="6" refX="1" refY="3" orient="auto">
              <path d="M6,0 L0,3 L6,6 Z" fill="#ef4444" />
            </marker>
          </defs>

          
          <text x="45" y="16" class="col-label" id="lbl-right-d">Domain</text>
          <text x="155" y="16" class="col-label" id="lbl-right-r">Range</text>

          
          <path id="arrow-4" d="M 62 52 L 138 80" class="map-arrow arrow-red" marker-end="url(#arr-r)" />
          <path id="arrow-5" d="M 62 112 L 138 84" class="map-arrow arrow-red" marker-end="url(#arr-r)" />

          
          <g transform="translate(45, 52)">
            <circle class="node-bg" r="16" />
            <text class="node-val">-2</text>
          </g>
          <g transform="translate(45, 112)">
            <circle class="node-bg" r="16" />
            <text class="node-val">2</text>
          </g>

          
          <g transform="translate(155, 82)">
            <circle class="node-bg" r="16" />
            <text class="node-val">4</text>
          </g>

          
          <g class="warning-cue" transform="translate(182, 82)">
            <text font-size="14" fill="#dc2626" font-weight="bold" dominant-baseline="central">?</text>
          </g>
        </svg>
      </div>

      <div class="badge-status badge-fail" id="badge-right">✗ Ambiguous reverse</div>
    </div>
  </div>

  <script>
    function setMode(mode) {
      const isReverse = mode === 'reverse';
      document.body.classList.toggle('reverse-mode', isReverse);

      document.getElementById('btnFwd').classList.toggle('active', !isReverse);
      document.getElementById('btnRev').classList.toggle('active', isReverse);

      // Markers toggle
      const markerGreen = isReverse ? 'url(#arr-g-rev)' : 'url(#arr-g)';
      const markerRed = isReverse ? 'url(#arr-r-rev)' : 'url(#arr-r)';

      ['arrow-1', 'arrow-2', 'arrow-3'].forEach(id => {
        const el = document.getElementById(id);
        if (isReverse) {
          el.removeAttribute('marker-end');
          el.setAttribute('marker-start', markerGreen);
        } else {
          el.removeAttribute('marker-start');
          el.setAttribute('marker-end', markerGreen);
        }
      });

      ['arrow-4', 'arrow-5'].forEach(id => {
        const el = document.getElementById(id);
        if (isReverse) {
          el.removeAttribute('marker-end');
          el.setAttribute('marker-start', markerRed);
        } else {
          el.removeAttribute('marker-start');
          el.setAttribute('marker-end', markerRed);
        }
      });

      // Update Labels
      document.getElementById('lbl-left-d').textContent = isReverse ? 'Output' : 'Domain';
      document.getElementById('lbl-left-r').textContent = isReverse ? 'Input' : 'Range';
      document.getElementById('lbl-right-d').textContent = isReverse ? 'Outputs' : 'Domain';
      document.getElementById('lbl-right-r').textContent = isReverse ? 'Input' : 'Range';

      document.getElementById('badge-left').textContent = isReverse ? '✓ 1 input → 1 output' : '✓ Reversible';
      document.getElementById('badge-right').textContent = isReverse ? '✗ 1 input → 2 outputs' : '✗ Ambiguous reverse';
    }
  </script>
</body>
</html>

OGM.12.FNC.3

Aligned learning resources

Related diagrams