## 1.4 Defining New Functions from Old

### Summary

Given a small number of basic functions, we can produce many more functions by stretching, shifting (both horizontally and vertically), and adding, just to mention a few of the possible operations on the basic set. In this section, we are going to study these and other standard ways to produce new functions from old ones, and the corresponding effects on the graphs.

By the end of your studying, you should know:

• How to identify various kinds of symmetry and reflections in graphs.
• How to scale a graph by modifying the function.
• How to shift a graph by modifying the function.
• How to define the sum, difference, product, and quotient of two functions (including the domains).
• How to define the composition of two functions (including the domain).
• How to determine if a function is one-to-one.
• How to determine if an inverse function exists.
• How to define the inverse of a function.

On-screen applet instructions: "Enter a function" means "type the function followed by the Enter key." Use the pull-down menu to see examples of how to indicate operands such as "*" for multiplication, "/" for division, etc.

### Examples

Consider the functions

and determine the domains of fog and gof.

Let

Find f –1(x), and then find the domains of f(x) and f–1(x).

Let f(x) be a function which contains the points

Determine its inverse.

### Applets

Stretching Graphs
Shifting Graphs
New Functions from Old
New Functions from Old Game
Arithmetical Operations on Functions
Inverse Functions

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 1.4 Defining New Functions from Old (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

Converting a drawing at a strange scale
to a standard scale for building models.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel