# Funky Functions

## Objective

In this lesson you will learn and understand how to use operations with functions. Below are examples and a video recap that will highlight the steps needs in order to successfully do operations on functions.

This can be found in the Accelerated Algebra 2 textbook, in chapter 6.1: Operations on Functions

## What is a Function?

A function is for every x there is only one y.

A function can be represented by an equation such as f(x)=2x+1

## Operations on Functions

• Subtract
• Multiply
• Divide
• Composition

Equation:

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

Steps:

1. Rewrite the functions f(x) and g(x) next to each other
2. Add together the two functions
3. (f+g)(x) is complete

Tip: When adding the functions together, only add the numbers that have the same degree, such as you can add 7x + -x together but not 7x + -1 since the -1 does not have x.

## Subtract Functions

Equation:

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

Steps:

1. Rewrite the functions f(x) and g(x) next to each other
2. Subtract the two functions together
3. (f-g)(x) is complete

Tip: When subtracting the functions together, only subtract the numbers that have the same degree, such as you can subtract -1 - 3 together but not 2x - 3 since the 3 does not have x.

## Multiply Functions

Equation:

(fg)(x) = f(x) g(x)

Steps:

1. Rewrite the functions f(x) and g(x) next to each other
2. Multiply together the two functions by distributing each letter or number in the first parenthesis to the second parenthesis.
3. Simply the multiplication of the two functions by either adding or subtracting the number and letter values together
4. (fg)(x) is complete

Tip: When multiplying the functions together, make sure to include all parts of the functions in fully multiplying through the two equations. Don't multiple similar values together such as (x+3)(x-2) = x^2-6, you have to multiply everything together such as (x+3)(x-2)= x^2-2x+3x-6. From there you would simplify by adding or subtracting similar degrees.

## Division Functions

Equation:

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

*g(x) cannot equal 0

Steps:

1. Rewrite the function f(x) above g(x)
2. Divide the two functions only if it can be or leave it as it is written
3. (f/g)(x) is complete

## Composition Functions

Equation:

f(g(x)) = (fog)(x)

Steps:

1. Rewrite the functions in which g(x) is written inside of f(x) - the x in the f(x) function will have the g(x) function within it
2. To simply f(g(x)) use arithmetic operations to get all the same degrees together
3. f(g(x)) is complete

Tip: When working on composition functions, make sure that all x values within f(x) have g(x) in them. For example: f(x) = x^2 + x and g(x) = x+5 which means f(g(x)) = (x+5)^2 +(x+5). From there follow factoring rules and arithmetic skills to gather all the same x's and number values together to reach the simplest form.