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What happens if you feed a number into an assembly line of math operations, and halfway through, a conveyor belt drops it off a cliff? When you chain equations together, valid inputs can suddenly turn into illegal mathematical operations like dividing by zero.
The domain of a function is the set of all real input values xx that produce a real output, while the range is the complete set of resulting output values yy.
What makes an input 'illegal' in precalculus? Let's inspect fractions where variables lurk in the basement.

Rational Functions

A rational function is a fraction of two polynomials, written as R(x)=P(x)Q(x)R(x) = \frac{P(x)}{Q(x)}. Because division by zero is undefined in real numbers, the domain excludes every xx-value where the denominator Q(x)=0Q(x) = 0.
Consider the function f(x)=2x+1x2−9f(x) = \frac{2x + 1}{x^2 - 9}. Setting the denominator x2−9=0x^2 - 9 = 0 yields (x−3)(x+3)=0(x - 3)(x + 3) = 0, meaning the domain is all real numbers except x=3x = 3 and x=−3x = -3, written in interval notation as (−∞,−3)∪(−3,3)∪(3,∞)(-\infty, -3) \cup (-3, 3) \cup (3, \infty).
To find the range analytically, solve the equation for xx in terms of yy. If y=2xx−5y = \frac{2x}{x - 5}, cross-multiplying gives y(x−5)=2xy(x - 5) = 2x, which rearranges to x(y−2)=5yx(y - 2) = 5y, or x=5yy−2x = \frac{5y}{y - 2}. Since y−2y - 2 cannot be zero, the range excludes 22: (−∞,2)∪(2,∞)(-\infty, 2) \cup (2, \infty).
A frequent pitfall is simplifying a fraction before checking its domain. For g(x)=x2−4x−2g(x) = \frac{x^2 - 4}{x - 2}, canceling (x−2)(x - 2) leaves x+2x + 2, but x=2x = 2 is still excluded from the domain because the original denominator was zero, creating a removable discontinuity (a hole) at (2,4)(2, 4).
What happens when the output of one function becomes the direct input of another? That requires passing through two security checkpoints.

Composite Functions

A composite function, written (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)), applies the inner function gg first, and then applies the outer function ff to that result.