You already know that \sqrt{9} = 3 because 3^2 = 9. The square root "undoes" the square. That undoing has a striking visual consequence: the graphs of y = x^2 and y = \sqrt{x} are mirror images of each other, reflected across the line y = x.

Seeing both curves on the same axes is one of the cleanest pictures of what it means for two operations to be inverses. One curve climbs steeply; the other curve grows with deliberate slowness. Fold the picture along the diagonal y = x and the two curves land exactly on top of each other.

The two curves, side by side

The square function y = x^2 turns small inputs into tiny outputs and blows up fast: 2^2 = 4, 3^2 = 9, 10^2 = 100. The square root function y = \sqrt{x} does the reverse — big inputs produce modest outputs: \sqrt{100} = 10, \sqrt{9} = 3, \sqrt{4} = 2.

Graphs of y equals x squared and y equals square root of x reflected across the line y equals x A square-shaped coordinate plane with both axes from zero to four. The parabola y equals x squared rises steeply from the origin. The square-root curve y equals root x rises slowly from the origin and curves off to the right. A dashed diagonal line y equals x runs from the origin to the top-right corner. The two curves are reflections of each other across this diagonal. Four key points are marked on each curve to show the reflection: one-one, two-four, three-nine-truncated, and the mirror points one-one, four-two, and nine-three-truncated. x y 1 2 3 4 1 2 3 4 y = x y = √x y = x² (1, 1) (4, 2) (2, 4) Fold the picture along the dashed line — the two curves land on top of each other.
Both curves pass through the origin and through $(1, 1)$. The parabola $y = x^2$ (green) climbs steeply upward — its slope keeps increasing. The square-root curve $y = \sqrt{x}$ (orange) climbs gently to the right — its slope keeps decreasing. The dashed line $y = x$ is the axis of reflection: every point on one curve has its mirror image on the other. For example, $(2, 4)$ on the parabola mirrors to $(4, 2)$ on the square root.

Look at how the two curves cross the line y = x: they both pass through the origin (0, 0) and through the point (1, 1). Those are the only two real fixed points of either function, because x^2 = x and \sqrt{x} = x both have solutions x = 0 and x = 1.

Why they are mirror images

Two functions f and g are inverses of each other when g(f(x)) = x and f(g(x)) = x — applying one undoes the other. For f(x) = x^2 and g(x) = \sqrt{x} with x \geq 0:

g(f(x)) = \sqrt{x^2} = x \quad \text{and} \quad f(g(x)) = (\sqrt{x})^2 = x

The general rule is that the graph of any function and its inverse are reflections of each other across the line y = x. Here is the reason.

If (a, b) is on the graph of f, that means f(a) = b. But if f and g are inverses, then g(b) = a — so the point (b, a) is on the graph of g. Swapping the coordinates of every point (a, b) \leftrightarrow (b, a) is exactly what reflection across the line y = x does geometrically. So the graph of g is the mirror image of the graph of f across that diagonal, automatically.

Why swapping coordinates is a reflection across y = x: the line y = x treats the two axes symmetrically. A point like (2, 4) sits above the line (because y > x); its mirror image (4, 2) sits below (because y < x). The reflection move "swap x and y" is the algebraic version of "flip across the diagonal." Every inverse-function pair is linked this way, not just square and square root.

A few specific mirror pairs

Check three points to feel the reflection:

Point on y = x^2 Mirror on y = \sqrt{x}
(1, 1) (1, 1) — on the diagonal, maps to itself
(2, 4) (4, 2)
(3, 9) (9, 3)
(0.5, 0.25) (0.25, 0.5)

Each row reads the same equation two ways. 3^2 = 9 says the parabola contains (3, 9); \sqrt{9} = 3 says the square-root curve contains (9, 3). Same fact, different graphs.

Why the parabola is restricted to x \geq 0 in this picture

The function y = x^2 accepts negative inputs too: (-3)^2 = 9, so the point (-3, 9) is also on the parabola. But the square-root function \sqrt{x} is only defined for x \geq 0 in the real numbers, and its output is always non-negative (by the principal-root convention from Roots and Radicals). So the clean mirror-image relationship only works in the first quadrant, where both curves live.

If you let the parabola extend to negative x, there is no matching square-root curve below the x-axis to reflect into — because \sqrt{x} is not defined for negative x, and its outputs are never negative. The mirror-image picture is a first-quadrant affair.

This is also why x^2 doesn't quite have a "full" inverse over all real numbers — it is not one-to-one, because 3^2 = (-3)^2 = 9 both give 9. The square-root function inverts only the right half of the parabola, the part with x \geq 0. To invert the left half, you would need the negative-root function y = -\sqrt{x}, whose graph is the reflection of the left half of the parabola across y = x.

The growth-rate contrast

Quantitatively, x^2 eventually dominates every polynomial-of-smaller-degree, while \sqrt{x} is dominated by every such polynomial. A doubling in x quadruples x^2 (factor of 4), but multiplies \sqrt{x} by only \sqrt{2} \approx 1.414 (factor of less than 1.5).

The two curves spread apart as fast as any pair of inverses can: one explodes, the other tames. That is the visual signature of a function and its inverse — they pull in exactly opposite directions, symmetrically across the diagonal.

The takeaway

y = x^2 and y = \sqrt{x} are the same equation read two different ways — the first says "given x, square it"; the second says "given y, find the x whose square is y." Plotted together, the two readings appear as reflections across the line y = x. That reflection is the universal visual signature of inverse functions: swap the coordinates, and any point on one curve lands on the other.

Related: Roots and Radicals · Exponents and Powers · Algebraic Identities · Quadratic Equations