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.
Answer
Attempt 2 out of 3

f, of, g, of, x, equalsf(g(x))=

g, of, f, of, x, equalsg(f(x))=

To determine if two functions are inverses of each other, we need to check the following using function composition:

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

Let’s start with the two functions:
- \( f(x) = 6x + 24 \)
- \( g(x) = \frac{1}{6}x - 4 \)

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

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

\[
f(g(x)) = f\left( \frac{1}{6}x - 4 \right)
\]
Now substitute \( \frac{1}{6}x - 4 \) for \( x \) in \( f(x) \):
\[
f\left( \frac{1}{6}x - 4 \right) = 6\left( \frac{1}{6}x - 4 \right) + 24
\]
Simplify:
\[
= x - 24 + 24 = x
\]
Thus, \( f(g(x)) = x \).

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

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

\[
g(f(x)) = g(6x + 24)
\]
Now substitute \( 6x + 24 \) for \( x \) in \( g(x) \):
\[
g(6x + 24) = \frac{1}{6}(6x + 24) - 4
\]
Simplify:
\[
= x + 4 - 4 = x
\]
Thus, \( g(f(x)) = x \).

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

Would you like any further details on function inverses, or have any questions about this process?

Here are 5 related questions:
1. How can you find the inverse of a function algebraically?
2. What are some common properties of inverse functions?
3. Can a function have more than one inverse?
4. What does the graph of an inverse function look like compared to the original?
5. How do you prove a function is not invertible?

**Tip**: To verify that two functions are inverses, both compositions \( f(g(x)) \) and \( g(f(x)) \) must equal \( x \).

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

Explore how to determine if two functions, f(x) = 6x + 24 and g(x) = (1/6)x - 4, are inverses of each other using function composition. This step-by-step guide covers the key algebraic concepts of inverse functions and their properties, suitable for students in Grades 9-11.

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