# Functions and Difference Quotients Review

I. Composition of Functions

II. Difference Quotients

Practice Problems

**I. Composition of Functions**

HAPPY BIRTHDAY... OK so in all likelihood today isn't your birthday.

Let's assume it is and you've been given a present wrapped in a large box.

You open the box and surprise!!!, inside is another box. Undaunted you

open it and discover yet another box. This may appear to be a lesson in ac-

cepting disappointment. Unbeknownst to you, though, you are experiencing

composite function behavior.

HUH?!?! Let us explain.

The composite function is like having one function
contained inside another.

When you see you probably think of it as just
another function,

but it's something more. It's a composite (one function inside another).

Here's why.

Suppose and g(x) =
2x−1 (think of as the larger

box). Now place 2x − 1 inside. Mathematically this is written as f(g(x))

and we call it "f composite g". To form a composite, try the following:

Now ll in the blanks with the g(x) representation and you
get

NOTE: just like f(3) means to input 3 into f, f(g(x))
means to input g

into f.

Unlike boxes, any function can be placed inside another.
From above,

g(x) = 2x −1. This really means g( )=2( )− 1. If
. we have

Alternative notation for composite functions:

Let's try a different practice drill. Suppose h(x) =
sin(2x). Can you state

functions for f and g so that h = f(g(x))?

Solution: inner function is 2x, which is g,

outer function is sin x = sin( ), which is f.

Check: f(g(x)) = f( )=sin( ).

Now insert 2x and get f(g(x)) = f(2x) = sin2x.

NOTE: This is the most obvious choice for f and g but not
the only

one. Let us try f(x) = sin(x + 1) and g(x) = 2x − 1.

Check: f(g(x)) = f( ) = sin[( ) + 1],

so f(g(x)) = f(2x − 1) = sin[(2x − 1) + 1] = sin 2x.

What's so special about composite functions? One day in
Calc I your

instructor will throw a birthday party for the entire class, and his present

to you will be called the CHAIN RULE.

**Exercise: Practice Problem 3.1.**

**II. Difference Quotients**

**A. Introduction**

In algebra, rate of change is introduced in its most basic
form by finding

the slope of a line using . This "formula"
can also be called

a Difference Quotient. In calculus, rates of change (both average and

instantaneous) are found using function forms of difference quotients.

Let's start with a few examples of the algebra in difference quotients.

**Example:** Let g(x) = x^{2} + 1. Evaluate and simplify
the difference

quotient

Solution: Using g(x) = x^{2} + 1 and g(2) = 5,

**Example:** Let Evaluate
and simplify the difference

quotient [ Δx represents the "change" in x.]

Remember:

**B. Application**

How does a difference quotient measure rate of change (or slope)?

Suppose we start with a function f(x) and draw a line (called a secant

line) connecting two of its points.

Notice that the slope is represented by a difference
quotient and is

referred to as the **average rate of change**. Now suppose the distance

between these two points is shortened by decreasing the size of h.

**Conclusion: **The two points get so close together they
almost con-

cide. The resulting line is now a tangent line whose slope measures the

instantaneous rate of change. You will come to know this slope by the

term "**derivative**".

PRACTICE PROBLEMS for Topic 3 - Functions and Difference Quotients

3.1 a) Suppose f(x) = x^{2} − 2x and
Form the following

compositions:

b) Suppose f(x) = sinx and Form the following compositions:

c) Suppose Select
functions for f, g, and h so that

y = f(g(h))(x). Check your results.

**Return to Review Topic Answers**

3.2 Find the rate of change (slope) of the line containing
the given points.

a) (2,−5) and (4,1)

b) (3,−4) and (3,−1)

c) (x, f(x)) and (x + h, f(x + h))

**Answers**

3.3 Evaluate and simplify the difference quotient
when:

a) f(x) = x^{2} − 3x,

b) (Rationalize the numerator).

**Answers**

3.4 Evaluate and simplify the difference quotient

when

**Answers**

ANSWERS to PRACTICE PROBLEMS

(Topic 3 - Functions and Difference Quotients)

c) inner: h(x) = 3x

middle: g(x) = sinx

outer:

CHECK:

**Return to Problem
**

3.2

a) m = 3

b) m is undefined

**Return to Problem**

**Return to Problem**

**Return to Problem**

**Beginning of Topic
150 Review Topics
Skills Assessment**