# Function Operation: Definition & Overview

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• 0:00 Functions
• 1:39 Function Operation
• 5:48 Lesson Summary

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Lesson Transcript
Instructor: Sarah Spitzig

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

In this lesson, we will define functions and determine how to use specific rules for the addition, subtraction, multiplication, and division of functions.

## Definition

A function is another way to think of an equation that has an x and a y value. We can think of x as the input value, or the value we plug into the equation to get the result. Likewise, we can think of y as the output value, or the result when we plug x into the equation.

For example, y = 3x, where x is the input and y is the output, in an equation. If we substituted 2 in for x as the input, than the output, or y, would be:

3(2) = 6.

Another example of an equation is y = -2x where x is the input and y is the output. If we substituted 5 in for x as the input, the output, or y, would be:

-2(5) = -10.

The only difference between an equation and a function is that instead of writing y as the output value, we write f(x).

The two equations y = 3x and y = -2x could be written as functions by changing the y to f(x).

f(x) = 3x is a function.

f'(x) = -2x is a function.

When we see f(x) = 2x - 3, this means that we want to find 2x - 3 for certain values of x. If I needed to find f(3), I would plug 3 into the equation as the input in order to get the output, so:

f(3) = 2x - 3
= 2(3) - 3
= 3

## Function Operations

Function operations are rules that we follow to solve functions. There is a certain way to deal with the addition, multiplication, and division of functions.

### Examples

• Function Addition Rule: If f(x) = 3x - 2, and g(x) = 4x -1, find (f + g)(5).

First, we use the function operation rules to find (f + g)(x) = f(x) + g(x). Then, we use the functions we were given, f(x) and g(x), and substitute = 3x - 2 + 4x - 1, which can be simplified to = 7x - 3 by grouping terms.

Then, we find (f + g)(5) by plugging 5 into our new function, so:

(f + g)(5) = -1(5) - 3
= -8

• Function Subtraction Rule: If f(x) = 3x - 2, and g(x) = 4x - 1, find (f- g)(5).

First, we use the operation rules to find (f - g)(x) = f(x) - g(x). Then, we use the functions we were given, f(x) and g(x), and substitute = 3x - 2 - (4x - 1), which can be simplified to -1x - 1 by grouping terms.

Then, we find (f- g)(5) by plugging 5 into our new function, so:

(f - g)(5) = -1(5) - 1
= -6

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