Announcements

Calculus IIMonday, January 30th

I WebAssign 2 due tonight!

I Problem Set 2 due Wednesday Feb 1

Today: Sec. 5.3: Inverse Functions

Understand when a function has an inverseWork with inverse functionsFind derivative of inverse at a point

Next Class: Sec. 5.4–5.5: Power Functions

Cherveny Jan 30 Math 1103: Calculus II

Warm-Up: Population Growth

Example

A bacteria population P is changing at rate

dP

dt=

3000

1 + .25t

where t is time in days. The initial population (t = 0) is 1000.Find the population after 3 days.

Answer:

P(3) = P(0) +

∫ 3

0P ′(x) dx = · · · = 1000(1 + 12 ln(1.75)) ≈ 7715

Cherveny Jan 30 Math 1103: Calculus II

One-to-One Functions

DefinitionA function is one-to-one if each output has a unique input.

Cherveny Jan 30 Math 1103: Calculus II

Inverse Functions

DefinitionIf f (x) is one-to-one, the inverse function f −1(x) sends eachoutput of f back to its unique input.

xf−→ f (x)

f −1

−→ x

In other words, f −1 is the function that “undoes” f .

I The range of f is the domain of f −1

I The domain of f is the range of f −1

Cherveny Jan 30 Math 1103: Calculus II

Graphing Inverse Functions

Graph of y = f −1(x) is the reflection of y = f (x) in the line y = x .

Recall: f (x) must pass the horizontal line test to have an inverse.

Cherveny Jan 30 Math 1103: Calculus II

Verifying Inverses

TheoremIf two functions f and g satisfy

f (g(x)) = x

g(f (x)) = x

then they are inverses of each other, i.e. f −1 = g and g−1 = f .

Cherveny Jan 30 Math 1103: Calculus II

Inverse Function Examples

Example

(a) Does f (x) = x2 have an inverse?

Answer: No (it fails the horizontal line test).

(b) Does f (x) = x2 on [0,∞) have an inverse?

Answer: Yes! Its inverse is f −1(x) =√x .

I Are the graphs reflections of each other in the line y = x?I Are the compositions what they should be?

Cherveny Jan 30 Math 1103: Calculus II

Calculating Inverses

Sometimes we can find f −1 by solving y = f (x) for x in terms of yand interchanging the variables.

Example

Suppose f (x) = 3x − 4. Does f −1(x) exist? Can you find it?

Answer:

f −1(x) =x + 4

3

Example

Let f (x) = (x + 2)2 + 3 on (−∞,−2]. What is f −1(x)?

Answer:f −1(x) = −2−

√x − 3

Cherveny Jan 30 Math 1103: Calculus II

Derivatives of Inverses

TheoremSuppose g(x) is the inverse of f (x). If f is differentiable,

g ′(x) =1

f ′(g(x))

provided f ′(g(x)) 6= 0

Proof.Use the geometrical properties of inverses.

Cherveny Jan 30 Math 1103: Calculus II

Practice

1. Verify that

f (x) =1

27(x5 + 2x3)

has an inverse and find (f −1)′(−11).

2. Are there any functions equal to their own inverse?

3. Let

f (x) =x + 6

x − 2x > 2

Find an inverse for f and calculate (f −1)′(2) by two methods.

Cherveny Jan 30 Math 1103: Calculus II

Practice Answers

1. Note that f ′(x) = 127(5x4 + 6x2) > 0 except at x = 0, so f is

one-to-one. Since f (−3) = −11, f −1(−11) = −3.

(f −1)′(−11) =1

f ′(f −1(−11))=

1

f ′(−3)=

1

17

2. Any function symmetric across the line y = x .

3.

f −1(x) =2x + 6

x − 1

Find that (f −1)′(2) = −8 by differentiating f −1(x) or byusing the formula for derivative of an inverse at a point.

Cherveny Jan 30 Math 1103: Calculus II