Cogito
Precalculus · Chapter 1 · Lesson 2
One-to-One Functions and Inverses
Only some functions can be undone.
12 problems · about 22 minutes · TEKS P.2.B, TEKS P.2.C
What this lesson teaches
The student tests a function for one-to-one behaviour and finds its inverse, restricting the domain when necessary.
- A function has an inverse exactly when it is one-to-one.
- The horizontal line test decides it from the graph.
- Restricting the domain can rescue a function that fails.
Warm Up
Straightforward practice. Get the method working first.
5 problems(2, 11) lies on f. What is f⁻¹(11)?
Answer 2
Why 2.
Why must a function be one-to-one to have an inverse?
Answer Otherwise one output would need to map back to two inputs.
Why The reverse would not be a function.
f(x) = 2x + 3. What is f⁻¹(9)?
Answer 3
Why (9 − 3) ÷ 2.
(5, 2) lies on f⁻¹. What is f(2)?
Answer 5
Why Swap the coordinates back.
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 problemsThe range of f is y ≥ 4. What is the smallest value in the domain of f⁻¹?
Answer 4
Why Domain and range trade places.
f(x) = x² on x ≥ 0. What is f⁻¹(49)?
Answer 7
Why The positive square root.
Which line does the graph of an inverse reflect across? Enter 1 for y = x, or 2 for the x-axis.
Answer 1
Why Swapping coordinates is a reflection across y = x.
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 − 7; f(x) = x³ + 1
Why Even powers and absolute values fold the number line in half.
f(x) = (x − 1) ÷ 4. What is f⁻¹(2)?
Answer 9
Why The inverse multiplies by 4 then adds 1.
The Point: The point (3, 8) lies on f. What is the x-coordinate of the matching point on f⁻¹?
Answer 8
Why 8, since (8, 3) lies on the inverse.
The Test: Which function fails the horizontal line test?
Answer f(x) = x²
Why x², because 3 and −3 both give 9.