# 45 Function Composition Common Core Algebra 2 Homework Answer Key

## Function Composition Common Core Algebra 2 Homework Answer Key

### Introduction

Function composition is an essential concept in algebra, particularly in the study of Common Core Algebra 2. It involves combining two or more functions to create a new function. This process can be challenging for students, especially when it comes to solving homework problems. In this article, we will provide a comprehensive answer key for common function composition problems in Common Core Algebra 2.

### Problem 1: Find the Composition of Two Functions

In this problem, you are given two functions, f(x) and g(x), and you need to find their composition, denoted as f(g(x)). To solve this, follow these steps:

- Replace x in f(x) with the expression g(x).
- Simplify the resulting expression.

Let's consider an example:

f(x) = 3x + 2

g(x) = 2x - 5

To find f(g(x)), we substitute g(x) into f(x):

f(g(x)) = 3(2x - 5) + 2

Simplifying further:

f(g(x)) = 6x - 15 + 2

f(g(x)) = 6x - 13

### Problem 2: Find the Inverse of a Composite Function

In this problem, you are given a composite function, and you need to find its inverse. To solve this, follow these steps:

- Write the composite function in the form f(g(x)).
- Swap the x and y variables in the composite function.
- Solve the resulting equation for y.
- The resulting equation represents the inverse of the composite function.

Let's consider an example:

f(g(x)) = 3x - 1

To find the inverse of f(g(x)), we swap x and y:

x = 3y - 1

Solving for y:

y = (x + 1) / 3

Therefore, the inverse of f(g(x)) is g^{-1}(x) = (x + 1) / 3.

### Problem 3: Evaluate a Composite Function

In this problem, you are given a composite function and a specific value of x, and you need to evaluate the function at that value. To solve this, follow these steps:

- Write the composite function in the form f(g(x)).
- Substitute the given value of x into the composite function.
- Simplify the resulting expression.

Let's consider an example:

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

To evaluate f(g(x)) at x = 4, we substitute 4 into the expression:

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

f(g(4)) = 2(16) - 12 + 1

f(g(4)) = 32 - 12 + 1

f(g(4)) = 21

### Problem 4: Solve an Equation Involving Composite Functions

In this problem, you are given an equation involving composite functions, and you need to solve for x. To solve this, follow these steps:

- Write the equation involving the composite functions.
- Isolate the composite function on one side of the equation.
- Find the inverse of the remaining function.
- Substitute the inverse function into the composite function.
- Solve the resulting equation for x.

Let's consider an example:

f(g(x)) = 5x - 2

To solve for x in the equation f(g(x)) = 5x - 2, we follow these steps:

Step 1: Isolate the composite function:

g(x) = f^{-1}(5x - 2)

Step 2: Find the inverse of the remaining function:

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

Step 3: Substitute the inverse function into the composite function:

g(x) = (5x - 2 + 2) / 5

Step 4: Simplify the expression:

g(x) = x

Step 5: Solve the resulting equation for x:

x = x

The solution to the equation is x = x, which means it is true for all real numbers.

### Problem 5: Graphing Composite Functions

In this problem, you are given two functions, f(x) and g(x), and you need to graph their composition, f(g(x)). To graph a composite function, follow these steps:

- Plot the points of the original functions, f(x) and g(x).
- Connect the points of f(x) and g(x) with a line.
- Label the points and the line as f(g(x)).

Let's consider an example:

f(x) = x^{2}

g(x) = 2x + 1

To graph f(g(x)), we plot the points of f(x) and g(x) and connect them with a line:

For f(x):

x = -2, y = 4

x = -1, y = 1

x = 0, y = 0

x = 1, y = 1

x = 2, y = 4

For g(x):

x = -2, y = -3

x = -1, y = -1

x = 0, y = 1

x = 1, y = 3

x = 2, y = 5

The points and the line represent the graph of f(g(x)).

### Problem 6: Find the Domain and Range of a Composite Function

In this problem, you are given a composite function, and you need to find its domain and range. To find the domain and range of a composite function, follow these steps:

- Identify the domains of the individual functions.
- Find the intersection of the domains.
- The resulting intersection represents the domain of the composite function.
- Identify the ranges of the individual functions.
- Find the range of the composite function by substituting the range values into the composite function.

Let's consider an example:

f(x) = x^{2}

g(x) = 2x + 1

To find the domain, we consider the individual domains:

Domain of f(x): All real numbers

Domain of g(x): All real numbers

The intersection of the domains is the set of all real numbers.

To find the range, we substitute the range values of f(x) into the composite function:

Range of f(x): [0, ∞)

Substituting the range values into f(g(x)):

f(g(x)) = (2x + 1)^{2}

Range of f(g(x)): [1, ∞)

Therefore, the domain of f(g(x)) is all real numbers, and the range is [1, ∞).

### Problem 7: Solve a Word Problem Involving Composite Functions

In this problem, you are given a real-life situation, and you need to model it using composite functions. To solve a word problem involving composite functions, follow these steps:

- Identify the variables and their relationships in the problem.
- Write an equation or a system of equations based on the given information.
- Represent the equations using composite functions.
- Solve the composite functions to find the solution to the problem.

Let's consider an example:

A car rental company charges a flat fee of $30 per day plus $0.25 per mile driven. Let x