Inverse

What is the notation for the inverse of a function f(x)?

f⁻¹(x)

Example: If f(x) = x + 4, what is f⁻¹(x)?

f⁻¹(x) = x − 4

What are the steps to find the inverse of a function?

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

Find the inverse of f(x) = 2x − 3

f⁻¹(x) = (x + 3)/2

What condition must a function satisfy to have an inverse?

It must be one-to-one (each input has a unique output)

What test helps determine if a function has an inverse?

Horizontal Line Test

Describe the Horizontal Line Test

If any horizontal line crosses the graph more than once, the function does NOT have an inverse

Example of a function that HAS an inverse

f(x) = 2x − 3 (straight sloped line)

Why does f(x) = 2x − 3 pass the Horizontal Line Test?

Every horizontal line crosses it at most once

Example of a function that does NOT have an inverse

f(x) = x² (U-shaped parabola)

Why does f(x) = x² fail the Horizontal Line Test?

A horizontal line like y = 4 crosses at x = −2 and x = 2, so it gives two outputs for one y