To verify that [tex]\( g(x) = \frac{1}{3}x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex], we need to show that [tex]\( g(f(x)) = x \)[/tex] and [tex]\( f(g(x)) = x \)[/tex].
So let's verify both conditions:
1. Verifying [tex]\( g(f(x)) = x \)[/tex]:
- First, apply [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 3x \][/tex]
- Next, apply [tex]\( g \)[/tex] to this result:
[tex]\[ g(f(x)) = g(3x) = \frac{1}{3}(3x) \][/tex]
- Simplify:
[tex]\[ g(f(x)) = x \][/tex]
This shows that applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex] results in [tex]\( x \)[/tex].
2. Verifying [tex]\( f(g(x)) = x \)[/tex]:
- First, apply [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{1}{3}x \][/tex]
- Next, apply [tex]\( f \)[/tex] to this result:
[tex]\[ f(g(x)) = f\left(\frac{1}{3}x\right) = 3\left(\frac{1}{3}x\right) \][/tex]
- Simplify:
[tex]\[ f(g(x)) = x \][/tex]
This shows that applying [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex] also results in [tex]\( x \)[/tex].
Therefore, we have verified that [tex]\( g(x) = \frac{1}{3}x \)[/tex] is indeed the inverse of [tex]\( f(x) = 3x \)[/tex].
Among the provided answer choices, the expression that correctly verifies this relationship is:
[tex]\[ \frac{1}{3}(3x) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{3}(3x)} \][/tex]
