Inverse functions

Undoing a function

An inverse function f⁻¹ reverses f. If f takes 3 to 7, then f⁻¹ takes 7 back to 3.

f(a) = b ⟺ f⁻¹(b) = a

Note: f⁻¹(x) is not 1/f(x). The −1 is notation, not an exponent.

Finding an inverse algebraically

1. Write y = f(x).
2. Swap x and y.
3. Solve for y.
4. Write y = f⁻¹(x).

Example: f(x) = 3x − 5

• y = 3x − 5
• x = 3y − 5
• y = (x + 5)/3
• f⁻¹(x) = (x + 5)/3

Check: f(f⁻¹(x)) = 3(x + 5)/3 − 5 = x ✓

Inverse from a table

Swap rows:

Graphs: reflection in y = x

The graph of f⁻¹ is the reflection of f in the line y = x. Every point (a, b) on f becomes (b, a) on f⁻¹.

• Domain of f = range of f⁻¹
• Range of f = domain of f⁻¹

When does an inverse exist?

Only one-to-one functions have inverse functions. Use the horizontal line test: if any horizontal line crosses the graph more than once, the inverse isn’t a function.

• y = 2x + 1: passes ✓
• y = x²: fails (both 2 and −2 give 4) ✗

Domain restriction

A domain restriction can fix this.

• f(x) = x², x ≥ 0 → f⁻¹(x) = √x

Common inverse pairs

Real-world inverses

• Converting currencies and back.
• Encoding and decoding messages.
• Temperature conversions.

Worked examples

Example 1: Inverse of f(x) = x/4 + 2? x = y/4 + 2 → f⁻¹(x) = 4(x − 2).

Example 2: f(5) = 12. What is f⁻¹(12)? 5.

Example 3: Does y = x³ have an inverse function? Passes the horizontal line test → yes, ∛x.

Common mistakes

• Treating f⁻¹(x) as 1/f(x).
• Forgetting to swap x and y.
• Ignoring domain restrictions for non-one-to-one functions.

Try it yourself

1. Inverse of f(x) = 2x + 7.
2. Inverse of f(x) = (x − 1)/3.
3. Inverse of f(x) = x³ + 2.
4. If (4, −1) is on f, which point is on f⁻¹?
5. Does y = |x| have an inverse function?

Answers: 1) (x − 7)/2 2) 3x + 1 3) ∛(x − 2) 4) (−1, 4) 5) not without a domain restriction

Quick recap

• An inverse function undoes f: swap x and y and solve.
• Graphs are reflections in y = x.
• Only one-to-one functions have inverses (horizontal line test).