Inverse Functions
Xiaoyue Cui
Department of Mathematics, University of Toronto
Oct 23,2025
Before next class:
Watch video 4.3,4.4 and complete
pre-class quiz 22
Download next class slides. No need to look at
them.
Learning Outcomes
After today’s lecture, you should be able to
1. Understand concepts related to functions (e.g.,
domain, codomain, range, one-to-one, onto,
invertible)
2. Determine whether a function is invertible and
compute its inverse
Warm up
A worm is crawling across the table. The path of the
worm looks something like this:
True or False?
The position of the worm is a function.
Worm function
A worm is crawling across the table.
For any time t, let f (t) be the position of the worm.
This defines a function f .
1. What is the domain of f ?
2. What is the codomain of f ?
3. What is the range of f ?
Fill in the Blank
Given that f is an invertible function, fill in the blanks.
1. If f (−1) = 0, then f −1 (0) = ——.
2. If f −1 (2) = 1, then f (1) = ——.
3. If (2, 3) is on the graph of f , then —— is on the
graph of f −1 .
4. If —— is on the graph of f , then (−2, 4) is on the
graph of f −1 .
Inverse function from a graph
Calculate:
1. f (2)
2. f (0)
3. f −1 (2)
4. f −1 (0)
5. f −1 (−1)
Finding a Restricted Domain on which a Function is Invertible
y
2
f
1
x
−2
−1
1
2
−1
1. Find the largest interval containing 0 on which f is
invertible.
2. Find the largest interval containing 1 on which f is
invertible.
Absolute value and inverses
Let
h(x) = x|x| + 1
1. Calculate h−1 (−8).
2. Sketch the graph of h.
3. Find an equation for h−1 .
4. Sketch the graph of h−1 .
5. Verify that
for every t ∈ ??? , h(h−1 (t)) = t.
for every t ∈ ??? , h−1 (h(t)) = t.
Composition of one-to-one functions
Assume for simplicity that all functions in this problem
have domain R. Prove the following theorem.
Theorem A
Let f and g be functions.
IF f and g are one-to-one,
THEN f ◦ g is one-to-one.
Suggestion:
1. Write the definition of what you want to prove.
2. Figure out the formal structure of the proof.
3. Complete the proof (use the hypotheses!)