https://file.aiquickdraw.com/mathbot/file/1723024650316i76ml0e3.jpg

The given functions are \( f(x) = 2x - 1 \) and \( g(x) = x^2 - 2x + 1 \). We need to find the following:

1. \( (f \circ g)(x) \)
2. \( (g \circ f)(x) \)
3. \( (f \circ f)(-1) \)
4. \( (g \circ g)(1) \)
5. \( (f \circ g)(1.5) \)

Let's solve these step by step:

### 1. \( (f \circ g)(x) \)
This means \( f(g(x)) \).

First, calculate \( g(x) \):
\[ g(x) = x^2 - 2x + 1 \]

Now, substitute \( g(x) \) into \( f \):
\[ f(g(x)) = f(x^2 - 2x + 1) \]
\[ f(x^2 - 2x + 1) = 2(x^2 - 2x + 1) - 1 \]
\[ = 2x^2 - 4x + 2 - 1 \]
\[ = 2x^2 - 4x + 1 \]

Thus,
\[ (f \circ g)(x) = 2x^2 - 4x + 1 \]

### 2. \( (g \circ f)(x) \)
This means \( g(f(x)) \).

First, calculate \( f(x) \):
\[ f(x) = 2x - 1 \]

Now, substitute \( f(x) \) into \( g \):
\[ g(f(x)) = g(2x - 1) \]
\[ g(2x - 1) = (2x - 1)^2 - 2(2x - 1) + 1 \]
\[ = (2x - 1)^2 - 4x + 2 + 1 \]
\[ = 4x^2 - 4x + 1 - 4x + 2 + 1 \]
\[ = 4x^2 - 8x + 4 \]

Thus,
\[ (g \circ f)(x) = 4x^2 - 8x + 4 \]

### 3. \( (f \circ f)(-1) \)
This means \( f(f(-1)) \).

First, calculate \( f(-1) \):
\[ f(-1) = 2(-1) - 1 \]
\[ = -2 - 1 \]
\[ = -3 \]

Now, substitute \(-3\) into \( f \):
\[ f(-3) = 2(-3) - 1 \]
\[ = -6 - 1 \]
\[ = -7 \]

Thus,
\[ (f \circ f)(-1) = -7 \]

### 4. \( (g \circ g)(1) \)
This means \( g(g(1)) \).

First, calculate \( g(1) \):
\[ g(1) = 1^2 - 2(1) + 1 \]
\[ = 1 - 2 + 1 \]
\[ = 0 \]

Now, substitute \( 0 \) into \( g \):
\[ g(0) = 0^2 - 2(0) + 1 \]
\[ = 0 + 0 + 1 \]
\[ = 1 \]

Thus,
\[ (g \circ g)(1) = 1 \]

### 5. \( (f \circ g)(1.5) \)
This means \( f(g(1.5)) \).

First, calculate \( g(1.5) \):
\[ g(1.5) = (1.5)^2 - 2(1.5) + 1 \]
\[ = 2.25 - 3 + 1 \]
\[ = 0.25 \]

Now, substitute \( 0.25 \) into \( f \):
\[ f(0.25) = 2(0.25) - 1 \]
\[ = 0.5 - 1 \]
\[ = -0.5 \]

Thus,
\[ (f \circ g)(1.5) = -0.5 \]

### Summary:
1. \( (f \circ g)(x) = 2x^2 - 4x + 1 \)
2. \( (g \circ f)(x) = 4x^2 - 8x + 4 \)
3. \( (f \circ f)(-1) = -7 \)
4. \( (g \circ g)(1) = 1 \)
5. \( (f \circ g)(1.5) = -0.5 \)

Do you want details or have any questions?

### Related Questions:
1. What is the derivative of \( (f \circ g)(x) \)?
2. What is the inverse of \( f(x) \)?
3. Find \( (g \circ f)(-2) \).
4. Calculate \( (f \circ f)(2) \).
5. Find the critical points of \( (g \circ g)(x) \).
6. What is \( (g \circ f)(3) \)?
7. Determine the range of \( (f \circ g)(x) \).
8. Calculate \( (f \circ g)(0) \).

### Tip:
When solving composite functions, always start by computing the inner function first and then substitute its result into the outer function.

Explore how to calculate composite functions (f ◦ g)(x) and (g ◦ f)(x) with detailed step-by-step solutions. Includes examples like (f ◦ g)(1.5) = -0.5 and (g ◦ f)(x) = 4x^2 - 8x + 4. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation