Let's analyze the given functions:
[tex]$f(x) = 11x \quad \text{and} \quad g(x) = \frac{x}{11}.$[/tex]
To determine if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other, we need to compute [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] and verify if both simplify to [tex]\( x \)[/tex].
### Part a: Calculate [tex]\( f(g(x)) \)[/tex]
1. Substitute [tex]\( g(x) = \frac{x}{11} \)[/tex] into [tex]\( f(x) = 11x \)[/tex].
[tex]\[
f(g(x)) = f\left( \frac{x}{11} \right).
\][/tex]
2. Apply the definition of [tex]\( f \)[/tex]:
[tex]\[
f\left( \frac{x}{11} \right) = 11 \left( \frac{x}{11} \right).
\][/tex]
3. Simplify the expression:
[tex]\[
11 \left( \frac{x}{11} \right) = x.
\][/tex]
Therefore, [tex]\( f(g(x)) = x \)[/tex].
### Part b: Calculate [tex]\( g(f(x)) \)[/tex]
1. Substitute [tex]\( f(x) = 11x \)[/tex] into [tex]\( g(x) = \frac{x}{11} \)[/tex].
[tex]\[
g(f(x)) = g(11x).
\][/tex]
2. Apply the definition of [tex]\( g \)[/tex]:
[tex]\[
g(11x) = \frac{11x}{11}.
\][/tex]
3. Simplify the expression:
[tex]\[
\frac{11x}{11} = x.
\][/tex]
Therefore, [tex]\( g(f(x)) = x \)[/tex].
### Part c: Determine if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses
To be inverses, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] must both simplify to [tex]\( x \)[/tex]. From parts a and b, we have:
[tex]\[
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x.
\][/tex]
Since both conditions are met, we can conclude that [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
Therefore, the two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
