https://file.aiquickdraw.com/mathbot/file/1729474853919tmgg5ly1.jpg

Let's go over each part of the problem in the image.

### Problem (a)

The functions are:
- \( f(x) = -\frac{4}{x} \), where \( x \neq 0 \)
- \( g(x) = -\frac{4}{x} \), where \( x \neq 0 \)

You are asked to find \( f(g(x)) \) and \( g(f(x)) \), and determine whether \( f \) and \( g \) are inverses.

#### Step 1: Find \( f(g(x)) \)
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(-\frac{4}{x}\right) = -\frac{4}{\left(-\frac{4}{x}\right)} = x
\]

#### Step 2: Find \( g(f(x)) \)
Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(-\frac{4}{x}\right) = -\frac{4}{\left(-\frac{4}{x}\right)} = x
\]

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are **inverses of each other**.

### Problem (b)

The functions are:
- \( f(x) = -x + 3 \)
- \( g(x) = x + 3 \)

#### Step 1: Find \( f(g(x)) \)
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x + 3) = -(x + 3) + 3 = -x - 3 + 3 = -x
\]

#### Step 2: Find \( g(f(x)) \)
Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(-x + 3) = (-x + 3) + 3 = -x + 6
\]

Since \( f(g(x)) = -x \) and \( g(f(x)) = -x + 6 \), the functions are **not inverses** of each other.

---

Would you like further details on any part of this? Here are 5 related questions to explore further:

1. How do you determine if two functions are inverses using their compositions?
2. What is the domain restriction for inverse functions, and why is it important?
3. Can a function have more than one inverse?
4. How does the algebraic process of simplifying compositions reveal inverse relationships?
5. What are common pitfalls in determining if two functions are inverses?

**Tip:** Inverse functions always "undo" each other, meaning their compositions \( f(g(x)) \) and \( g(f(x)) \) must equal \( x \).

For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other.

Learn how to determine if two functions, f(x) and g(x), are inverses of each other by evaluating their compositions, f(g(x)) and g(f(x)). This problem explores core algebra concepts like function composition and inverse functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation