https://file.aiquickdraw.com/mathbot/file/1729474253848y7auso6t.jpg

The problem asks about the composition of two functions, \( f(x) \) and \( g(x) \), whose graphs are given in the image. Let's break down how to solve the two parts of the question.

The composition of functions, such as \( (g \circ f)(x) \), means that you first apply the function \( f(x) \), and then apply the function \( g(x) \) to the result.

### 1. Part (a): \( (g \circ f)(x) \)

This is calculated as:

\[
(g \circ f)(x) = g(f(x))
\]

You first find \( f(x) \) from the graph of \( f \), and then use that value to find \( g(f(x)) \) from the graph of \( g \).

Let's solve it step-by-step for different values of \( x \):

#### \( x = 1 \):
- From the graph of \( f(x) \), \( f(1) = 1 \).
- Now, use this value in the graph of \( g \). \( g(1) = 3 \).
- Therefore, \( (g \circ f)(1) = g(f(1)) = 3 \).

#### \( x = 2 \):
- From the graph of \( f(x) \), \( f(2) = 2 \).
- Now, \( g(2) = 4 \).
- Therefore, \( (g \circ f)(2) = g(f(2)) = 4 \).

#### \( x = 3 \):
- From the graph of \( f(x) \), \( f(3) = 1 \).
- Now, \( g(1) = 3 \) (as before).
- Therefore, \( (g \circ f)(3) = g(f(3)) = 3 \).

#### \( x = 4 \):
- From the graph of \( f(x) \), \( f(4) = 4 \).
- Now, \( g(4) = 2 \).
- Therefore, \( (g \circ f)(4) = g(f(4)) = 2 \).

So the values for \( (g \circ f)(x) \) are:
\[
(g \circ f)(1) = 3, \quad (g \circ f)(2) = 4, \quad (g \circ f)(3) = 3, \quad (g \circ f)(4) = 2.
\]

### 2. Part (b): \( (f \circ g)(x) \)

This is calculated as:

\[
(f \circ g)(x) = f(g(x))
\]

Now, we will use \( g(x) \) first and then apply \( f(x) \).

#### \( x = 1 \):
- From the graph of \( g(x) \), \( g(1) = 3 \).
- Now, from the graph of \( f(x) \), \( f(3) = 1 \).
- Therefore, \( (f \circ g)(1) = f(g(1)) = 1 \).

#### \( x = 2 \):
- From the graph of \( g(x) \), \( g(2) = 4 \).
- Now, \( f(4) = 4 \).
- Therefore, \( (f \circ g)(2) = f(g(2)) = 4 \).

#### \( x = 3 \):
- From the graph of \( g(x) \), \( g(3) = 2 \).
- Now, \( f(2) = 2 \).
- Therefore, \( (f \circ g)(3) = f(g(3)) = 2 \).

#### \( x = 4 \):
- From the graph of \( g(x) \), \( g(4) = 1 \).
- Now, \( f(1) = 1 \).
- Therefore, \( (f \circ g)(4) = f(g(4)) = 1 \).

So the values for \( (f \circ g)(x) \) are:
\[
(f \circ g)(1) = 1, \quad (f \circ g)(2) = 4, \quad (f \circ g)(3) = 2, \quad (f \circ g)(4) = 1.
\]

Would you like more details on any of the steps? Here are some related questions for further practice:

1. What is the value of \( (f \circ f)(x) \) for different values of \( x \)?
2. What is the value of \( (g \circ g)(x) \)?
3. How would you find the inverse of \( f(x) \) and \( g(x) \) based on the graphs?
4. Can you generalize a formula for \( (g \circ f)(x) \) if \( f(x) \) and \( g(x) \) are given as functions?
5. What is the domain and range of \( (g \circ f)(x) \) and \( (f \circ g)(x) \)?

**Tip:** When working with function compositions, it's helpful to break down each step and focus on applying one function at a time.

The graphs of f(x) = y and g(x) = y are given below. What is the value of (g o f)(x) and (f o g)(x)?

Learn how to solve a function composition problem using graphs of f(x) and g(x). This exercise covers key concepts in function composition, focusing on how to apply one function to the result of another. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation