To verify that [tex]\( g(x) = \frac{1}{3} x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex], we need to check if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex].
1. First verification: [tex]\( f(g(x)) = x \)[/tex]
Given [tex]\( g(x) = \frac{1}{3} x \)[/tex]:
[tex]\[
f(g(x)) = f\left(\frac{1}{3} x\right)
\][/tex]
Now, apply the function [tex]\( f \)[/tex] to [tex]\( \frac{1}{3} x \)[/tex]:
[tex]\[
f\left(\frac{1}{3} x\right) = 3 \left(\frac{1}{3} x\right)
\][/tex]
Simplify the expression:
[tex]\[
3 \left(\frac{1}{3} x\right) = x
\][/tex]
So, [tex]\( f(g(x)) = x \)[/tex] is satisfied.
2. Second verification: [tex]\( g(f(x)) = x \)[/tex]
Given [tex]\( f(x) = 3x \)[/tex]:
[tex]\[
g(f(x)) = g(3x)
\][/tex]
Now, apply the function [tex]\( g \)[/tex] to [tex]\( 3x \)[/tex]:
[tex]\[
g(3x) = \frac{1}{3} (3x)
\][/tex]
Simplify the expression:
[tex]\[
\frac{1}{3} (3x) = x
\][/tex]
So, [tex]\( g(f(x)) = x \)[/tex] is also satisfied.
Since both conditions [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] are satisfied, [tex]\( g(x) = \frac{1}{3} x \)[/tex] is indeed the inverse of [tex]\( f(x) = 3x \)[/tex].
Now, let's match this with the given options to determine which expression could be used to verify the inverse relationship:
1. [tex]\( 3 x\left(\frac{x}{3}\right) \)[/tex] - This reads as [tex]\( 3x \cdot \frac{x}{3} \)[/tex], simplifying to [tex]\( x^2 \)[/tex], which does not match our needs.
2. [tex]\( \left(\frac{1}{3} x\right)(3 x) \)[/tex] - This reads as [tex]\( \left(\frac{1}{3} x\right) \cdot 3 x \)[/tex], simplifying to [tex]\( x^2 \)[/tex], which does not match our needs.
3. [tex]\( \frac{1}{3}(3 x) \)[/tex] - This reads as [tex]\( \frac{1}{3} \cdot 3x \)[/tex], simplifying to [tex]\( x \)[/tex], which matches our needs.
4. [tex]\( \frac{1}{3}\left(\frac{1}{3} x\right) \)[/tex] - This reads as [tex]\( \frac{1}{3} \cdot \frac{1}{3} x \)[/tex], simplifying to [tex]\( \frac{1}{9} x \)[/tex], which does not match our needs.
Therefore, the correct expression to verify that [tex]\( g(x) = \frac{1}{3} x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex] is:
[tex]\[
\boxed{\frac{1}{3}(3 x)}
\][/tex]
