Operations on Functions Lesson

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.

Polynomial Functions

A polynomial function of degree n (largest exponent is n) is defined by:
f(x) = a_nxⁿ + a_n-1x^{n - 1} + ... + a₁x¹ + a₀
a_n, a_{n - 1}, ..., a₀ are real numbers, n is a whole number, and a_n is not zero.
A few examples of polynomial functions:
f(x) = 9x² - 4x + 1
h(x) = -2x³ - 5
g(x) = 14x⁶ + 12x² + 7
In some cases, we want to evaluate a function for a given value of the variable. Let's take our function:
f(x) = 9x² - 4x + 1
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) = 9x² - 4x + 1
f(2) = 9(2)² - 4(2) + 1
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)² - 4(-3) + 1
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) = 2x³ - 5x + 11
g(-1) = 2(-1)³ - 5(-1) + 11
g(-1) = 14
g(7) = 2(7)³ - 5(7) + 11
g(7) = 2(343) - 35 + 11
g(7) = 662
g(3) = 2(3)³ - 5(3) + 11
g(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) = 17x² - 3x - 5
g(x) = 11x² + x - 1
(f + g)(x) = (17x² - 3x - 5) + (11x² + x - 1)
(f + g)(x) = 17x² + 11x² - 3x + x - 5 - 1
(f + g)(x) = 28x² - 2x - 6
(f - g)(x) = (17x² - 3x - 5) - (11x² + x - 1)
(f - g)(x) = 17x² - 11x² - 3x - x - 5 + 1
(f - g)(x) =  6x² - 4x - 4
Example 3: Find (f + g)(-3)
f(x) = 5x³ - 7x + 13
g(x) = -3x² - 10x + 1
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) = (5x³ - 7x + 13) + (-3x² - 10x + 1)
f(x) + g(x) = 5x³ - 3x² - 17x + 14
Now we can replace each x with a -3:
(f + g)(-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
f(-3) = -101
g(-3) = -3(-3)² - 10(-3) + 1
g(-3) = 4
(f + 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) = 2x² - 3x - 17
g(x) = 9x² + x - 21
f(-8) = 2(-8)² - 3(-8) - 17
f(-8) = 135
g(-8) = 9(-8)² - 8 - 21
g(-8) = 547
f(-8) • g(-8) = 135 • 547 = 73,845
(fg)(-8) = 73,845
Example 5: Find (f/g)(-2)
f(x) = 3x² - 5x + 1
g(x) = 7x - 9
f(-2) = 3(-2)² - 5(-2) + 1
f(-2) = 23
g(-2) = 7(-2) - 9
g(-2) = -23
(f/g)(-2) = (23)/(-23) = -1
(f/g)(-2) = -1

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