Precalculus

Inverse Functions

An inverse function undoes what the original function does. If f takes 3 to 9, then f-inverse takes 9 back to 3. It’s like having an “undo” button for math. But not every function has an inverse — and understanding why is just as important as knowing how.

1. What Does “Inverse” Mean Graphically?

If you have a point (a, b) on the graph of f, then the point (b, a) is on the graph of f-inverse. Swapping x and y coordinates means the inverse is a reflection over the line y = x.

-2-112345678-2-112345678f(x) = x² (x >= 0)f⁻¹(x) = sqrt(x)y = x

The purple curve (x squared, restricted to x >= 0) and the red curve (square root) are mirror images across the gray line y = x. That reflection relationship is the visual signature of inverse functions.

2. Exploring Inverses with a Slider

Let’s look at the linear function f(x) = mx + b and its inverse. A linear function always has an inverse (as long as m is not zero). Adjust the slope and intercept and watch both the function and its inverse update together.

Slope (m)2
-33
Intercept (b)1
-55
f(x)=2x+1f(x) = 2x + 1
f−1(x)=x−12f^{-1}(x) = \frac{x - 1}2
-8-6-4-22468-8-6-4-22468f(x) = mx + bf⁻¹(x)y = x
Try This

Watch how the function and its inverse are always symmetric about y = x. Try setting m = 1 — the function and inverse become parallel lines on opposite sides of y = x. What happens when m = -1? The function is its own inverse! (It reflects onto itself.)

3. The Horizontal Line Test

Not every function has an inverse. For a function to be invertible, it must be one-to-one: every output comes from exactly one input. The visual test? If any horizontal line crosses the graph more than once, the function fails.

Horizontal line y = ?4
-510
-4-224-224681012y = x² (NOT one-to-one)y = h (test line)
Connection

Slide the horizontal line up and down. For any positive y-value, the line hits the parabola in two places (e.g., both x = 2 and x = -2 give y = 4). That’s why x-squared doesn’t have an inverse over all real numbers. To fix this, we restrict the domain to x >= 0 (or x <= 0), keeping only the half where it’s one-to-one.

4. Power Function and Its Inverse

Let’s explore f(x) = x^n and its inverse (the nth root). Adjust the exponent and see how the function and its inverse relate. We’ll restrict to x >= 0 to keep things invertible.

Exponent (n)2
15
f(x)=x2,f−1(x)=x1/2f(x) = x^2, \quad f^{-1}(x) = x^{1/2}
-1123456-1123456f(x) = x^nf⁻¹(x) = x^(1/n)y = x
Try This

When n = 1, the function is just f(x) = x, which is its own inverse. As n increases, the function curve bends more sharply away from y = x, and so does the inverse (in the opposite direction). At n = 2, you get the square/square-root pair. At n = 3, the cube/cube-root pair.

5. Verifying: f(f-inverse(x)) = x

The defining property of inverse functions: if you compose a function with its inverse, you get back the identity function (a straight line through the origin with slope 1). Let’s verify this visually.

Stretch (a)2
0.54
f(x)=2⋅x2,f−1(x)=x/2f(x) = 2 \cdot x^2, \quad f^{-1}(x) = \sqrt{x / 2}
f(f−1(x))=2⋅(x/2)2=xf(f^{-1}(x)) = 2 \cdot \left(\sqrt{x/2}\right)^2 = x
-112345678-112345678f(x) = a*x²f⁻¹(x) = sqrt(x/a)f(f⁻¹(x)) = xy = x (identity)
Challenge

Challenge: No matter what value of a you pick, the teal curve (the composition) always lands exactly on the gray identity line y = x. Change the slider to convince yourself. This is the algebraic definition of an inverse: f(f-inverse(x)) = x AND f-inverse(f(x)) = x. Both directions must hold.
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