https://file.aiquickdraw.com/mathbot/file/1726683470695lurgw8rg.jpg

Let's work through the problems step by step for both parts (a) and (b).

### Part (a)
We are given:
- \( f(x) = \frac{-6}{x}, x \neq 0 \)
- \( g(x) = \frac{-6}{x}, x \neq 0 \)

We need to check if \( f \) and \( g \) are inverses of each other by calculating:
1. \( f(g(x)) \)
2. \( g(f(x)) \)

**Step 1: Calculate \( f(g(x)) \)**

Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left( \frac{-6}{x} \right) = \frac{-6}{\frac{-6}{x}} = x
\]
Thus, \( f(g(x)) = x \).

**Step 2: Calculate \( g(f(x)) \)**

Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left( \frac{-6}{x} \right) = \frac{-6}{\frac{-6}{x}} = x
\]
Thus, \( g(f(x)) = x \).

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

### Part (b)
We are given:
- \( f(x) = -x + 2 \)
- \( g(x) = x + 2 \)

Again, we need to check if \( f \) and \( g \) are inverses by calculating:
1. \( f(g(x)) \)
2. \( g(f(x)) \)

**Step 1: Calculate \( f(g(x)) \)**

Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x + 2) = -(x + 2) + 2 = -x - 2 + 2 = -x
\]
Thus, \( f(g(x)) = -x \).

**Step 2: Calculate \( g(f(x)) \)**

Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(-x + 2) = (-x + 2) + 2 = -x + 4
\]
Thus, \( g(f(x)) = -x + 4 \).

Since \( f(g(x)) \neq x \) and \( g(f(x)) \neq x \), \( f \) and \( g \) are **not** inverses of each other.

### Final Answers:
- **(a)** \( f(g(x)) = x \), \( g(f(x)) = x \). Therefore, \( f \) and \( g \) **are** inverses of each other.
- **(b)** \( f(g(x)) = -x \), \( g(f(x)) = -x + 4 \). Therefore, \( f \) and \( g \) **are not** inverses of each other.

---

Do you have any questions or need further details on any step? Here are five related questions you might want to explore:

1. What are the conditions for two functions to be inverses of each other?
2. How can you determine the domain of a function?
3. Can two functions be inverses if their ranges are restricted to a particular domain?
4. What happens if the domain of a function is not specified?
5. How can you verify if a composition of functions results in the identity function?

**Tip:** When calculating function compositions, always substitute carefully to avoid sign or algebraic errors.

For each pair of functions f and g, find f(g(x)) and g(f(x)), then determine whether f and g are inverses of each other. The given functions are: (a) f(x) = -6/x, g(x) = -6/x and (b) f(x) = -x + 2, g(x) = x + 2.

Learn how to determine if two functions are inverses of each other by calculating their compositions. This problem involves algebraic functions and covers key concepts such as function composition and inverse functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation