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Chapter 3

Function Basics

A function is a special type of rule or relationship. The difference between a function and a relation is that a function has exactly one output value (from the range) for every input value (from the domain). Functions are very useful when you’re describing trends in business, heights of objects shot from a cannon, times required to complete a task, and so on. Functions have some special properties and operations that allow for investigation into what happens when you change the rule.

The Problems You’ll Work On

In this chapter, you’ll work with functions and function operations in the following ways:

 Writing and using function notation

 Determining the domain and range of different types of functions

 Recognizing even and odd functions

 Checking on whether a function is one-to-one

 Finding inverses of one-to-one functions

 Performing the basic operations on functions and function rules

 Working with the composition of functions and the difference quotient

What to Watch Out For

Don’t let common mistakes trip you up; keep in mind that when working with functions, your challenges will include

 Following the order of operations when evaluating functions

 Determining which values need to be excluded from a function’s domain

 Working with negative signs correctly when checking for even and odd functions

 Being sure a function is one-to-one before trying to determine an inverse

 Correctly applying function rules when performing function composition

 Raising binomials to higher powers and including all the terms

Using Function Notation to Evaluate Function Values

121−125 Evaluate the function for the given value.

121. Given , find .

122. Given , find .

123. Given , find .

124. Given , find .

125. Given , find .

Determining the Domain and Range of a Function

126−135 Find the domain and range for the function.

126.

127.

Illustration by Thomson Digital

128.

129.

130.

131.

132.

133.

134.

135.

Recognizing Even Functions

136−137 Determine which function is even.

136. , , , ,

137. , , , ,

Identifying Odd Functions

138−139 Determine which function is odd.

138. , , , ,

139. , , , ,

Ruling Out Even and Odd Functions

140 Determine which function is neither even nor odd.

140. , , , ,

Recognizing One-to-One Functions from Given Relations

141−143 Determine which of the given relations is a one-to-one function over its domain.

141. , ,

142. ,,,,

143.

Illustration by Thomson Digital

Illustration by Thomson Digital

Illustration by Thomson Digital

Illustration by Thomson Digital

Illustration by Thomson Digital

Identifying One-to-One Functions from Equations

144−145 Determine which of the functions is one-to-one over its domain.

144. , , , ,

145. , , , ,

Recognizing a Function’s Inverse

146−150 Determine which pair of functions are inverses of each other.

146. ,,,,

147. ,,,,

148. , ,,

149. , ,,,

150. ,,,

Determining a Function’s Inverse

151−160 Find the inverse of the function.

151.

152.

153.

154.

155.

156.

157.

158.

159.

160.

Executing Operations on Functions

161−165 Perform the indicated operation using the given functions.

161.

Find .

162.

Find .

163.

Find .

164.

Find .

165.

Find .

Performing Function Composition

166−169 Find .

166.

167.

168.

169.

Doing More Function Composition

170−176 Perform the indicated operation.

170.

Find .

171.

Find .

172.

Find .

173.

Find .

174.

Find .

175.

Find .

176.

Find .

Using the Difference Quotient

177−180 Evaluate the difference quotient, , for the given function Assume .

177.

178.

179.

180.