Given the functions f, of, x, equals, 6, x, plus, 24f(x)=6x+24 and g, of, x, equals, one sixth, x, minus, 4g(x)= 
6
1
​
 x−4 determine if the two functions are inverses by using composition.

To determine if two functions \( f(x) \) and \( g(x) \) are inverses of each other, we can use composition. Specifically, we need to check two conditions:

1. \( (f \circ g)(x) = f(g(x)) = x \)
2. \( (g \circ f)(x) = g(f(x)) = x \)

Let’s apply these steps.

### Given:
- \( f(x) = 6x + 24 \)
- \( g(x) = \frac{1}{6}x - 4 \)

#### Step 1: Check \( f(g(x)) \)
Substitute \( g(x) = \frac{1}{6}x - 4 \) into \( f(x) \):

\[
f(g(x)) = f\left(\frac{1}{6}x - 4\right) = 6\left(\frac{1}{6}x - 4\right) + 24
\]
Now simplify:

\[
f(g(x)) = 6 \cdot \frac{1}{6}x - 6 \cdot 4 + 24 = x - 24 + 24 = x
\]

So, \( f(g(x)) = x \).

#### Step 2: Check \( g(f(x)) \)
Substitute \( f(x) = 6x + 24 \) into \( g(x) \):

\[
g(f(x)) = g(6x + 24) = \frac{1}{6}(6x + 24) - 4
\]
Now simplify:

\[
g(f(x)) = \frac{1}{6} \cdot 6x + \frac{1}{6} \cdot 24 - 4 = x + 4 - 4 = x
\]

So, \( g(f(x)) = x \).

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

---

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the general method for finding the inverse of a function?
2. How do you verify inverses using graphical representation?
3. What happens if \( f(g(x)) \neq x \) in composition?
4. Can all functions have inverses? Why or why not?
5. How does domain and range affect the existence of inverses?

**Tip:** When working with inverses, always check both compositions \( f(g(x)) \) and \( g(f(x)) \) to ensure the functions are truly inverses.

Given the functions f(x)=6x+24 and g(x)=1/6x-4, determine if the two functions are inverses by using composition.

Learn how to verify if f(x)=6x+24 and g(x)=1/6x-4 are inverse functions through composition. This detailed step-by-step solution uses the properties of function composition and inverse functions. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation