Pre-Calculus Chapter 1

Functions and Their Graphs

Warm Up 1.6 A high-altitude spherical weather

balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at a constant rate of 0.03 inches per second and that r = 48 inches at time t = 0. Determine an equation that models the volume V of the balloon at time t and find the volume when t = 300 seconds.Note: Volume of a sphere .

2

3

3

4rV

1.6 Inverse FunctionsObjectives:

Find inverse functions numerically and algebraically.

Verify that two functions are inverse functions of each other.

Determine if functions are one-to-one. Use graphs of functions to decide

whether functions have inverse functions.

3

Consider This …

A highway crew is painting the center line on a road. Knowing the crew’s previous work record, it is possible to predict how much of the stripe the crew will have painted at any time during a normal 8-hour shift. It may also be possible to tell how long the crew has been working by how much stripe has been painted.

4

Mathematically …

For the first case, the distance is dependent on time.

For the second, time is dependent on the distance.

5

)(tfd

)(dgt

Inverses If a new relation is formed by

interchanging the input and output variables in a given relation, the two relations are inverses of each other.

If both relations are functions, they are called inverse functions.

Notation: The inverse of f (x) is f -1(x).

6

Inverse of a Function Numerically Let d be the number of

miles a highway crew paints in an 8-hour shift.

Let t be the number of hours the crew has been on the job.

Find f (1), f (2), etc. Find g(1), g(2), etc. Inputs & outputs are

interchanged.7

t (hours)

d (miles)

1 0.2

2 0.6

3 1.0

4 1.4

5 1.8

6 2.2

7 2.6

8 3.0

)(tfd

)(dgt

Graphs of the Functions

8

x

y

x

y

d = f (t) t = g(d)

Inverse of a Function Graphically

When a function is graphed on the same set of axes as its inverse, we see that the function and its inverse are reflections across the line y = x.

9

x

y

Inverse of a Function Algebraically

To find the inverse of a function

algebraically:

Exchange x-values and y-values.

Solve for y.

Call the original function f (x) and

the inverse function f -1(x).

10

Algebraic Inverse Example

11

Linear function for

the function f.

or

Interchange

variables.

Solve for y in terms

of x.

Label function as f

(x) and inverse as f -

1(x).

)5.0(4.0 xy

2.04.0 xy

2.04.0 yx

5.05.2 xy

2.04.0)( xxf

5.05.2)(1 xxf

Composition of Inverse Functions

The composition of a function f with its

inverse f -1 results in the identity function.

The functions f and f -1 “undo” one another.

That is, and

.

12

xxff ))(( 1 xxff ))((1

Domain and Range of Inverse Functions

If we interchange the x-values and the y-values to create the inverse of a function, it must follow that: The domain of f is the range of f -1 and The range of f is the domain of f -1.

Likewise, The domain of f -1 is the range of f and The range of f -1 is the domain of f .

13

Existence of an Inverse

How do we know that a relation is a function? It passes the Vertical Line Test. Each input has only one output.

For the function to have an inverse, Each output must have only one input.

The function must be one-to-one.14

One-to-One Functions A one-to-one function has exactly

one y for each x and exactly one x for each y.

If a function is one-to-one, it will pass the Horizontal Line Test.

A function has an inverse if and only if it is one-to-one.

If f is increasing on its entire domain or if it is decreasing on its entire domain, then f is one-to-one and, therefore, has an inverse.

15

Example 1 Given .a. Make a table of values for f (–2), f (–1), f (0), f

(1), and f (2). From the numbers in the table, explain why you cannot find the value of x if f (x) = 2.5. How does this result indicate that we cannot find an inverse for f ?

b. Plot the five points for f and connect with a smooth curve. Do the same for the inverse relation (on the same set of axes). How does the graph confirm that the inverse function for f cannot be found?

c. Plot the line y = x on the same set of axes. What do you see?

16

25.0)( 2 xxf

Homework 1.6 Worksheet 1.6# 6, 7, 9, 11, 17, 20, 21 – 24 (matching), 27,

32, 34, 39, 56, 57, 61, 62, 67, 68, 83, 84

17