To verify that [tex]\( g(x) = \frac{1}{3} x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex], we need to verify the compositions [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex]:
1. Finding [tex]\( f(g(x)) \)[/tex]:
[tex]\[
\begin{aligned}
g(x) &= \frac{1}{3} x, \\
f(g(x)) &= f\left( \frac{1}{3} x \right).
\end{aligned}
\][/tex]
Since [tex]\( f(x) = 3x \)[/tex]:
[tex]\[
f\left( \frac{1}{3} x \right) = 3 \left( \frac{1}{3} x \right).
\][/tex]
Simplify the expression inside the function:
[tex]\[
3 \left( \frac{1}{3} x \right) = x.
\][/tex]
So, [tex]\( f(g(x)) = x \)[/tex].
2. Finding [tex]\( g(f(x)) \)[/tex]:
[tex]\[
\begin{aligned}
f(x) &= 3x, \\
g(f(x)) &= g(3x).
\end{aligned}
\][/tex]
Since [tex]\( g(x) = \frac{1}{3} x \)[/tex]:
[tex]\[
g(3x) = \frac{1}{3} (3x).
\][/tex]
Simplify the expression inside the function:
[tex]\[
\frac{1}{3} (3x) = x.
\][/tex]
So, [tex]\( g(f(x)) = x \)[/tex].
Both compositions [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] hold true, confirming that [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].
Now let's match the choices given with our findings:
1. [tex]\( 3 x \left( \frac{x}{3} \right) \)[/tex]
This translates to:
[tex]\[
3 x \left( \frac{x}{3} \right) = x \cdot x = x^2,
\][/tex]
which is not what we are looking for.
2. [tex]\( \left( \frac{1}{3} x \right)(3 x) \)[/tex]
This translates to:
[tex]\[
\left( \frac{1}{3} x \right)(3 x) = \frac{1}{3} x \cdot 3 x = x^2,
\][/tex]
which is also not correct.
3. [tex]\( \frac{1}{3}(3 x) \)[/tex]
This translates to:
[tex]\[
\frac{1}{3} (3 x) = x,
\][/tex]
which matches one of our findings and verifies the inverse relationship.
4. [tex]\( \frac{1}{3} \left( \frac{1}{3} \pi \right) \)[/tex]
This is not relevant to the problem, as it uses [tex]\(\pi\)[/tex] which is not part of our original functions.
The correct expression that verifies [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[
\boxed{\frac{1}{3}(3 x)}
\][/tex]
