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Verify Functions
Are Inverses

Learn to prove two functions are inverses by composing them and checking that both compositions simplify to \( x \).

Algebra 2

What Is an Inverse Function?

An inverse function "undoes" what the original function does. If \( f \) takes an input \( x \) and produces output \( y \), then \( f^{-1} \) takes \( y \) back to \( x \).

\( f \) : Forward
\( f(x) = 2x + 3 \) takes \( 4 \) to \( 11 \)
\( f^{-1} \) : Reverse
\( f^{-1}(x) = \frac{x-3}{2} \) takes \( 11 \) back to \( 4 \)

The Composition Test

Instead of graphing or guessing, we verify inverse functions algebraically by composing them. The key idea: if you apply \( f \) then \( f^{-1} \) (or vice versa), you should get back the original input \( x \).

This is the only rigorous algebraic method to confirm two functions are inverses: it works for all values in the domain, not just a few test points.

The Round-Trip Visual

\(x\)
\(f\)
\(f(x)\)
\(f^{-1}\)
\(x\)
\(= x\)
\(f^{-1}\bigl(f(x)\bigr) = x\)

The Inverse Verification Theorem

When both compositions equal \( x \), we write \( g = f^{-1} \). You must check both directions: one composition equaling \( x \) alone is not sufficient in general.

Graphs Reflect Over \( y = x \)

\(y=x\)
\(f(x) = 2x + 1\)
\(g(x) = \frac{x-1}{2}\)
 x y 1 2 3 4 5 1 2 3

Verifying vs. Finding Inverses

Verifying Inverses Finding Inverses
Given Both \( f \) and \( g \) are provided Only \( f \) is provided
Goal Confirm \( f \) and \( g \) are inverses Derive \( f^{-1} \) from scratch
Method Compose: check both equal \( x \) Swap \( x \) and \( y \), solve for \( y \)
Result Yes or No A new function formula

Step-by-Step Verification

1
Identify \( f(x) \) and \( g(x) \). Write both function rules clearly.
2
Compute \( f(g(x)) \). Substitute the entire expression for \( g(x) \) into every \( x \) in \( f \). Simplify completely.
3
Compute \( g(f(x)) \). Substitute the entire expression for \( f(x) \) into every \( x \) in \( g \). Simplify completely.
4
Check both results. If \( f(g(x)) = x \) AND \( g(f(x)) = x \), they are inverses. Otherwise, they are not.

Example 1: Linear Functions

Verify that \( f(x) = 3x - 5 \) and \( g(x) = \dfrac{x + 5}{3} \) are inverses.

Example 2: Rational Functions

Verify that \( f(x) = \dfrac{x}{x - 1} \) and \( g(x) = \dfrac{x}{x - 1} \) are inverses of each other (for \( x \neq 1 \)).

Example 3: Cube Root Pair

Verify: \( f(x) = x^3 + 2 \) and \( g(x) = \sqrt[3]{x - 2} \).

When They Are Not Inverses

Show that \( f(x) = x^2 \) and \( g(x) = x - 1 \) are not inverses.

Example 4: Square Root Pair

Verify: \( f(x) = x^2 - 4 \) (for \( x \geq 0 \)) and \( g(x) = \sqrt{x + 4} \).

Common Mistakes to Avoid

✕ Only Checking One Direction
Showing \( f(g(x)) = x \) but skipping \( g(f(x)) \). Both must equal \( x \) for a complete proof.
✕ Substitution Errors
Replacing only some instances of \( x \), or forgetting to apply the outer function to the entire inner expression.
✕ Ignoring Domain Restrictions
For functions like \( x^2 \) and \( \sqrt{x} \), forgetting that the domain must be restricted for the inverse to exist.

Summary of Inverse Pairs

\( f(x) \) \( f^{-1}(x) \) Key Operation
\( 3x - 5 \) \( \frac{x+5}{3} \) Undo multiply, then subtract
\( x^3 + 2 \) \( \sqrt[3]{x - 2} \) Undo cube, then add
\( x^2 - 4,\ x \geq 0 \) \( \sqrt{x + 4} \) Undo square (restricted domain)
\( \frac{x}{x-1} \) \( \frac{x}{x-1} \) Self-inverse (involution)

Question 1 of 5

To verify that \( f \) and \( g \) are inverses, you must show that:

Question 2 of 5

Let \( f(x) = 4x + 7 \) and \( g(x) = \dfrac{x - 7}{4} \). What is \( f(g(x)) \)?

Question 3 of 5

If \( f(x) = x^3 - 1 \) and \( g(x) = \sqrt[3]{x + 1} \), what is \( g(f(x)) \)?

Question 4 of 5

Are \( f(x) = 2x + 3 \) and \( g(x) = \dfrac{x - 2}{3} \) inverses?

Question 5 of 5

Why must we restrict the domain of \( f(x) = x^2 \) to \( x \geq 0 \) before it can have an inverse?