AI Tutor
If you square 44, you get 1616. But if you square −4-4, you also get 1616.
If two different inputs lead to the exact same output, how can you run the process backward to land on a single definite answer?

The Horizontal Line Test

A relation is a function only if every input produces exactly one output.
The horizontal line test tells us whether an inverse will be a function: if any horizontal line crosses a graph more than once, reflecting that graph across the diagonal line y=xy = x makes it fail the vertical line test.
To solve this problem, we apply a domain restriction by slicing the parabola at its turning point (the vertex) and keeping just one half, typically x≥0x \ge 0.
With only the right half active, each output pairs with exactly one input, making an inverse function possible.
So what does the algebraic inverse of that restricted curve actually look like?

Inverting the Parent Quadratic

Start with the parent quadratic f(x)=x2f(x) = x^2 restricted to x≥0x \ge 0, and rewrite it as y=x2y = x^2.
Swap the roles of input and output to get x=y2x = y^2, then take the square root of both sides to isolate yy.
Because the original domain was x≥0x \ge 0, our new output must satisfy y≥0y \ge 0, which selects the principal positive root: f−1(x)=xf^{-1}(x) = \sqrt{x}.