Concept 8

Solve problems involving inverse relationships or inverse functions when presented.

Inverse functions reverse the original input-output rule.

Core Idea

An inverse function undoes the original function. To find 𝑓−1(𝑎), ask: what input to 𝑓 would produce the output 𝑎?

Understanding

Rule: ACT inverse-function questions are usually about reversing a rule, not memorizing a new formula. If 𝑓(𝑥) =3𝑥 −7, then 𝑓−1(11) means the number that gets sent to 11 by 𝑓.

That is why solving 3𝑥 −7 =11 works. You are tracing the function backward from output to input. The same idea appears in inverse variation and real-world relationships where one quantity increases as another decreases.

Step by Step

  1. Replace 𝑓(𝑥) with the given output value.
  2. Solve for the original input.
  3. Check by substituting your answer back into the original function.
  4. Keep input and output roles straight: the inverse reverses them.
Question

Worked Example

If 𝑓(𝑥) =3𝑥 −7, what is 𝑓−1(11)?

Select an answer to see the explanation