 determine   whether   each   pair   of   functions are inverses for g(x)=4x+1 and f(x)=-4x-1

To determine whether the given pair of functions are inverses, we need to check if:

- \( f(g(x)) = x \)
- \( g(f(x)) = x \)

Given:
- \( g(x) = 4x + 1 \)
- \( f(x) = -4x - 1 \)

### Step 1: Check \( f(g(x)) \)

Substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(4x + 1) = -4(4x + 1) - 1
\]

Simplify:

\[
f(g(x)) = -16x - 4 - 1 = -16x - 5
\]

Since \( f(g(x)) \neq x \), the functions are **not inverses** based on this.

### Step 2: Check \( g(f(x)) \)

Substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(-4x - 1) = 4(-4x - 1) + 1
\]

Simplify:

\[
g(f(x)) = -16x - 4 + 1 = -16x - 3
\]

Since \( g(f(x)) \neq x \), the functions are also **not inverses** based on this check.

### Conclusion:
The functions \( g(x) = 4x + 1 \) and \( f(x) = -4x - 1 \) are **not inverses** of each other.

Would you like more details on any step or further explanation?

Here are 5 related questions to expand your understanding:
1. How do you algebraically verify if two functions are inverses?
2. What is the geometric interpretation of inverse functions on a graph?
3. Can you find the inverse of \( g(x) = 4x + 1 \)?
4. How can you check for inverse functions using compositions of functions?
5. What happens to the slopes of linear functions when they are inverses?

**Tip:** When finding the inverse of a function, swap \( x \) and \( y \) and then solve for \( y \) to get the inverse.

Determine whether each pair of functions are inverses for g(x)=4x+1 and f(x)=-4x-1.

This detailed solution explains how to determine if two functions, g(x) = 4x + 1 and f(x) = -4x - 1, are inverses by using function composition. Suitable for students in Grades 10-12, the solution walks through both f(g(x)) and g(f(x)) calculations, demonstrating the process for verifying 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