1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 2: Functions and Graphs
Please review this lecture (from MATH 1100 class) before you begin the section 5.7 (Inverse Trigonometric functions)
2.6 Combinations of Functions; Composite Functions
2.7 Inverse Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Find the domain of a function.
Combine functions using the algebra of functions, specifying domains.
Form composite functions.
Determine domains for composite functions.
Write functions as compositions.

3. Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.

4. Example: Finding the Domain of a Function
Find the domain of the function
Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.
We exclude 7 and – 7 from the domain of g.
The domain of g is

5. The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotient
are functions whose domains are the set of all real numbers common to the domains of f and
defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:

6. Example: Combining Functions
Let and Find each of the following: a.
b. The domain of
The domain of f(x) has no restrictions.
The domain of g(x) has no restrictions.
The domain of is

7. The Composition of Functions
The composition of the function f with g is denoted
and is defined by the equation
The domain of the composite function is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.

8. Example: Forming Composite Functions
Given and find

9. Excluding Values from the Domain of
The following values must be excluded from the
input x:
If x is not in the domain of g, it must not be in the domain of
Any x for which g(x) is not in the domain of f must not be in the domain of

10. Example: Forming a Composite Function and Finding Its Domain
Given and
Find

11. Example: Forming a Composite Function and Finding Its Domain
Given and
Find the domain of
For g(x),
For
The domain of is

12. Example: Writing a Function as a Composition
Express h(x) as a composition of two functions:
If and then

13. Objectives:
Verify inverse functions.
Find the inverse of a function.
Use the horizontal line test to determine if a function has an inverse function.
Use the graph of a one-to-one function to graph its inverse function.
Find the inverse of a function and graph both functions on the same axes.

14. Definition of the Inverse of a Function
Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g
and
g(f(x)) = x for every x in the domain of f
The function g is the inverse of the function f and is denoted f –1 (read “f-inverse). Thus, f(f –1 (x)) = x and
f –1(f(x))=x. The domain of f is equal to the range of
f –1, and vice versa.

15. Example: Verifying Inverse Functions
Show that each function is the inverse of the other:
and
verifies that f and g are inverse functions.

16. Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
1. Replace f(x) with y in the equation for f(x).
2. Interchange x and y.
3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

17. Finding the Inverse of a Function (continued)
The equation for the inverse of a function f can be found as follows:
4. If f has an inverse function, replace y in step 3 by
f –1(x). We can verify our result by showing that
f(f –1 (x)) = x and f –1 (f(x)) = x

18. Example: Finding the Inverse of a Function
Find the inverse of
Step 1 Replace f(x) with y:
Step 2 Interchange x and y:
Step 3 Solve for y:
Step 4 Replace y with f –1 (x):

19. The Horizontal Line Test for Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

20. Example: Applying the Horizontal Line Test
Which of the following graphs represent functions that have inverse functions?
a. b.
Graph b represents a function that has an inverse.

21. Graphs of f and f – 1
The graph of f –1 is a reflection of the graph of f about the line y = x.

22. Example: Graphing the Inverse Function
Use the graph of f to draw the graph of f –1

23. Example: Graphing the Inverse Function (continued)
We verify our solution by observing the reflection of the graph about the line y = x.