# Composition of Functions Worksheets

There are situations where we can put two or more functions together to form a new function. This operation is referred to as composition of functions. The basic concept is very fundamental all that you need to do is replace the variable (x) of the function on the left with the function on the right and process it through. We use this procedure often when we are calculating measure unit conversions and financial functions such as sale or reduced percentage pricing. This series of worksheets and lessons will help students understand how to combine together and interact with multiple functions.

### Aligned Standard: HSF-BF.A.1c

- Function Composition Step-by-step Lesson- We split the problem into inner and outer functions to play around with.
- Guided Lesson - We work on finding the function rule. This really baffles kids at first.
- Guided Lesson Explanation - This is mostly a plug and chug set of problems. Question two does require a little more work.
- Practice Worksheet - Evaluate the composition of all the mixed functions. They all follow a similar pattern.
- Matching Worksheet - I tried to throw a few extra setups with exponents in here.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

These sheets are all about managing functions and finding the outcome of their relation.

- Homework 1 - Evaluate the inner function. Insert the answer from step 1 into outer function and evaluate further.
- Homework 2 - Plug m – 2 into f(c) and simplify.
- Homework 3 - The two functions t(x) and v(x) are defined below. t(x) = 6x – 4 v(x) = x
^{2}+ 5

### Practice Worksheets

It's time for two functions to come together. Tell us how the marriage works out.

- Practice 1 - Evaluate the composition of functions v(t(6))
- Practice 2 - Use the defined functions to solve the end game.
- Practice 3 - Find the final value of all of the terms you are given first.

### Math Skill Quizzes

Evaluate all of the composition of these functions.

- Quiz 1 - Use the following function rule to find f(m -2). Simplify your answer. F(c) = 7c
- Quiz 2 - Finish this quiz all the way through.
- Quiz 3 - Use everything you learned to pull this one together.

### What is Function Composition?

Function composition in mathematics refers to an operation that uses two functions f ( ) and g ( ) and, as a result, produces another function. In essence we function composition is the act of putting two or more functions into a single function. The result composition is written as (g ° f)(x) which means g(f(x)). As you can see here, we are substituting f(x) for the x in g(x). The core idea with this concept is that the input of the function is not a numeric value, but instead is another function.

Let's consider an example : f(x) = 5x + 7 and g(x)= x^{2}

Not to forget that "x" is only to denote the unknown formula, Here's the first step;

f(x)= 5x+7, g(x)= (x)^{2}

We first use f then apply for the result;

(g x f)(x) = (5x + 7)^{2}

Always be careful about which function comes first because if you inverse the function, the answer also gets affected. It is also important to note that like most of math, the order matters. Functions are composed from right to left. When you are given two functions, the one on the right always gets placed inside the one located on the left.

You might remember doing something very similar to this in a measurement unit. When we are converting units of measure, we are actually using this same skill. Say you know how many miles per hour something was travelling and then you want to apply this to a physics problem that notes speed in meters per second, this is standard function composition to make that conversion. Physics is just riddled with all types of problems like this.