Cogito
Integrated Math 3 · Chapter 3 · Lesson 3
Inverse Functions
Undoing a function, when it can be undone.
12 problems · about 22 minutes · F-BF.B.4
What this lesson teaches
The student finds inverse functions, restricts domains where needed, and verifies by composition.
- An inverse exists exactly when the function is one-to-one.
- Swap x and y, then solve for y.
- Verify by composing in both directions.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = 2x + 3. What is f⁻¹(9)?
Answer 3
Why 3.
Why must a function be one-to-one to have an inverse?
Answer Otherwise one output would have to map back to two inputs.
Why The reverse would not be a function.
f(x) = x + 5. What is f⁻¹(12)?
Answer 7
Why Subtract 5.
f(x) = 4x. What is f⁻¹(20)?
Answer 5
Why Divide by 4.
f(x) = x³. What is f⁻¹(27)?
Answer 3
Why The cube root.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = x² on x ≥ 0. What is f⁻¹(49)?
Answer 7
Why The positive root.
f(f⁻¹(9)) equals what?
Answer 9
Why They undo each other.
The range of f is y ≥ 4. Smallest value in the domain of f⁻¹?
Answer 4
Why Domain and range swap.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich functions are one-to-one over all real numbers?
Answer f(x) = 3x minus 7; f(x) = x cubed plus 1
Why Even powers and absolute values fold the number line.
f(x) = (x − 1) ÷ 4. What is f⁻¹(2)?
Answer 9
Why Multiply by 4 then add 1.
The Inverse: f(x) = 3x + 6. What is f⁻¹(12)?
Answer 2
Why 2.
The Point: The point (3, 8) lies on f. What is f⁻¹(8)?
Answer 3
Why 3.