Lesson Objectives

- Demonstrate an understanding of function notation
- Learn how to add/subtract polynomial functions
- Learn how to multiply/divide polynomial functions

## Operations on Functions

Many lessons ago, we introduced the concept of a function. Additionally, we
learned how to write functions using function notation. In this lesson, we will learn how
to perform operations (addition, subtraction, multiplication, and division) with polynomial functions.

f(x) = a

a

A few examples of polynomial functions:

f(x) = 9x

h(x) = -2x

g(x) = 14x

In some cases, we want to evaluate a function for a given value of the variable. Let's take our function:

f(x) = 9x

Suppose we want to know the functions value when x is 2. We can show this by replacing the x inside of the parentheses with a 2. We will then replace each x in our function with a 2 and simplify:

f(x) = 9x

f(2) = 9(2)

f(2) = 29

This tells us the functions value is 29, when the variable x is 2.

What if we wanted to find the functions value when x is -3?

f(-3) = 9(-3)

f(-3) = 94

This tells us the functions value is 94, when the variable is -3.

Let's look at an example.

Example 1: Find g(-1), g(7), g(3)

g(x) = 2x

g(-1) = 2(-1)

g(-1) = 14

g(7) = 2(7)

g(7) = 2(343) - 35 + 11

g(7) = 662

g(3) = 2(3)

g(3) = 50

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

Let's look at a few examples.

Example 2: Find (f + g)(x) and (f - g)(x)

f(x) = 17x

g(x) = 11x

(f + g)(x):

(17x

17x

(f + g)(x) = 28x

(f - g)(x):

(17x

17x

6x

(f - g)(x) = 6x

Example 3: Find (f + g)(-3)

f(x) = 5x

g(x) = -3x

For this scenario, we have two options.

1) find f(x) + g(x) and substitute in a -3 for x in the result.

2) find f(-3) + g(-3).

Let's start with the first method.

f(x) + g(x):

(5x

5x

Now we can replace each x with a -3:

5(-3)

-97

(f + g)(-3) = -97

Now let's use the second method.

f(-3):

5(-3)

-101

g(-3):

-3x

-3(-3)

4

f(-3) + g(-3) = -101 + 4 = -97

Either way, our answer is -97.

(fg)(x) = f(x) • g(x)

(f/g)(x) = f(x)/g(x)

Let's look at a few examples.

Example 4: Find (fg)(-8)

f(x) = 2x

g(x) = 9x

f(-8) = 2(-8)

f(-8) = 135

g(-8) = 9(-8)

g(-8) = 547

f(-8) • g(-8) = 135 • 547 = 73,845

(fg)(-8) = 73,845

Example 5: Find (f/g)(-2)

f(x) = 3x

g(x) = 7x - 9

f(-2) = 3(-2)

f(-2) = 23

g(-2) = 7(-2) - 9

g(-2) = -23

(f/g)(-2) = (23)/(-23) = -1

(f/g)(-2) = -1

### Polynomial Functions

A polynomial function of degree n (largest exponent is n) is defined by:f(x) = a

_{n}x^{n}+ a_{n-1}x^{n - 1}+ ... + a_{1}x^{1}+ a_{0}a

_{n}, a_{n - 1}, ..., a_{0}are real numbers, n is a whole number, and a_{n}is not zero.A few examples of polynomial functions:

f(x) = 9x

^{2}- 4x + 1h(x) = -2x

^{3}- 5g(x) = 14x

^{6}+ 12x^{2}+ 7In some cases, we want to evaluate a function for a given value of the variable. Let's take our function:

f(x) = 9x

^{2}- 4x + 1Suppose we want to know the functions value when x is 2. We can show this by replacing the x inside of the parentheses with a 2. We will then replace each x in our function with a 2 and simplify:

f(x) = 9x

^{2}- 4x + 1f(2) = 9(2)

^{2}- 4(2) + 1f(2) = 29

This tells us the functions value is 29, when the variable x is 2.

What if we wanted to find the functions value when x is -3?

f(-3) = 9(-3)

^{2}- 4(-3) + 1f(-3) = 94

This tells us the functions value is 94, when the variable is -3.

Let's look at an example.

Example 1: Find g(-1), g(7), g(3)

g(x) = 2x

^{3}- 5x + 11g(-1) = 2(-1)

^{3}- 5(-1) + 11g(-1) = 14

g(7) = 2(7)

^{3}- 5(7) + 11g(7) = 2(343) - 35 + 11

g(7) = 662

g(3) = 2(3)

^{3}- 5(3) + 11g(3) = 50

### Adding and Subtracting Polynomial Functions

In some cases, we will need to add or subtract functions. If f(x) and g(x) define functions, then:(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

Let's look at a few examples.

Example 2: Find (f + g)(x) and (f - g)(x)

f(x) = 17x

^{2}- 3x - 5g(x) = 11x

^{2}+ x - 1(f + g)(x):

(17x

^{2}- 3x - 5) + (11x^{2}+ x - 1)17x

^{2}+ 11x^{2}- 3x + x - 5 - 1(f + g)(x) = 28x

^{2}- 2x - 6(f - g)(x):

(17x

^{2}- 3x - 5) - (11x^{2}+ x - 1)17x

^{2}- 3x - 5 - 11x^{2}- x + 16x

^{2}- 4x - 4(f - g)(x) = 6x

^{2}- 4x - 4Example 3: Find (f + g)(-3)

f(x) = 5x

^{3}- 7x + 13g(x) = -3x

^{2}- 10x + 1For this scenario, we have two options.

1) find f(x) + g(x) and substitute in a -3 for x in the result.

2) find f(-3) + g(-3).

Let's start with the first method.

f(x) + g(x):

(5x

^{3}- 7x + 13) + (-3x^{2}- 10x + 1)5x

^{3}- 3x^{2}- 17x + 14Now we can replace each x with a -3:

5(-3)

^{3}- 3(-3)^{2}- 17(-3) + 14-97

(f + g)(-3) = -97

Now let's use the second method.

f(-3):

5(-3)

^{3}- 7(-3) + 13-101

g(-3):

-3x

^{2}- 10x + 1-3(-3)

^{2}- 10(-3) + 14

f(-3) + g(-3) = -101 + 4 = -97

Either way, our answer is -97.

### Multiplying and Dividing Polynomial Functions

Additionally, we will need to multiply or divide functions. If f(x) and g(x) define functions, then:(fg)(x) = f(x) • g(x)

(f/g)(x) = f(x)/g(x)

Let's look at a few examples.

Example 4: Find (fg)(-8)

f(x) = 2x

^{2}- 3x - 17g(x) = 9x

^{2}+ x - 21f(-8) = 2(-8)

^{2}- 3(-8) - 17f(-8) = 135

g(-8) = 9(-8)

^{2}- 8 - 21g(-8) = 547

f(-8) • g(-8) = 135 • 547 = 73,845

(fg)(-8) = 73,845

Example 5: Find (f/g)(-2)

f(x) = 3x

^{2}- 5x + 1g(x) = 7x - 9

f(-2) = 3(-2)

^{2}- 5(-2) + 1f(-2) = 23

g(-2) = 7(-2) - 9

g(-2) = -23

(f/g)(-2) = (23)/(-23) = -1

(f/g)(-2) = -1

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