Skip to content
Mom's Basement University

Inverse Functions

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll review inverse functions and extend your understanding to more complex cases, including how to find inverses algebraically and graphically, and when they exist.

The Concept

Section titled “The Concept”

If you went through the Algebra 2 section, you’ve already seen inverse functions. This lesson is a recap of the basics plus some new territory. In Algebra 2 we focused on finding inverses of linear functions and understanding the concept. Here we’ll go further: inverses of rational functions, domain restrictions on non-one-to-one functions, and formal verification using composition.

Two functions f and g are inverses if applying one after the other returns the original input:

f(g(x))=xandg(f(x))=xf(g(x)) = x \quad \text{and} \quad g(f(x)) = x

The inverse of f is written f⁻¹ (read “f inverse”). It undoes whatever f does. If f takes 3 to 7, then f⁻¹ takes 7 back to 3.

Graphically, the inverse is the reflection of the original function over the line y = x. Every point (a, b) on f becomes (b, a) on f⁻¹.

To find the inverse algebraically:

  1. Replace f(x) with y
  2. Swap x and y
  3. Solve for y
  4. Replace y with f⁻¹(x)

Not every function has an inverse. A function must be one-to-one (pass the horizontal line test) to have an inverse. If it’s not one-to-one, you can restrict the domain to make it work.

Worked Example

Section titled “Worked Example”

Example 1: Linear function

Find the inverse of f(x) = 3x - 6.

y=3x−6y = 3x - 6

Swap x and y:

x=3y−6x = 3y - 6

Solve for y:

x+6=3y⇒y=x+63x + 6 = 3y \quad \Rightarrow \quad y = \frac{x + 6}{3} f−1(x)=x+63f^{-1}(x) = \frac{x + 6}{3}

Verify they are inverses:

f(f−1(x))=3(x+63)−6=(x+6)−6=x✓f(f^{-1}(x)) = 3\left(\frac{x + 6}{3}\right) - 6 = (x + 6) - 6 = x \quad \checkmark f−1(f(x))=(3x−6)+63=3x3=x✓f^{-1}(f(x)) = \frac{(3x - 6) + 6}{3} = \frac{3x}{3} = x \quad \checkmark

Both compositions return x, so they’re confirmed inverses.

Example 2: Quadratic with restricted domain

Find the inverse of f(x) = x² + 4, with domain restricted to x ≥ 0.

y=x2+4y = x^2 + 4

Swap x and y:

x=y2+4x = y^2 + 4

Solve for y:

y2=x−4⇒y=x−4y^2 = x - 4 \quad \Rightarrow \quad y = \sqrt{x - 4}

We take the positive root because the original domain was x ≥ 0.

f−1(x)=x−4,x≥4f^{-1}(x) = \sqrt{x - 4}, \quad x \geq 4

Example 3: Rational function

Find the inverse of f(x) = (2x + 1) / (x - 3).

y=2x+1x−3y = \frac{2x + 1}{x - 3}

Swap and solve:

x=2y+1y−3x = \frac{2y + 1}{y - 3} x(y−3)=2y+1x(y - 3) = 2y + 1 xy−3x=2y+1xy - 3x = 2y + 1 xy−2y=3x+1xy - 2y = 3x + 1 y(x−2)=3x+1y(x - 2) = 3x + 1 f−1(x)=3x+1x−2f^{-1}(x) = \frac{3x + 1}{x - 2}

Real-World Application

Section titled “Real-World Application”

Inverse functions are used in:

  • Unit conversion (Fahrenheit to Celsius and back: C = (5/9)(F - 32) and F = (9/5)C + 32)
  • Cryptography (encoding and decoding messages)
  • Finance (finding the interest rate given a final amount, or the original price before tax)
  • Physics (converting between position, velocity, and time)
  • Computer science (compression and decompression algorithms)

Example: if a store charges f(x) = 1.08x (price with 8% tax), the inverse f⁻¹(x) = x / 1.08 tells you the pre-tax price from the total.

Quiz

Section titled “Quiz”
Two functions $f$ and $g$ are inverses if:
The graph of $f^{-1}$ is the reflection of $f$ over:
A function has an inverse only if it is:
If $f(x) = 4x - 12$, then $f^{-1}(x)$ is:
The inverse of $f(x) = x^2 + 1$ (with $x \geq 0$) is: