To verify if [tex]\( g(x) = \frac{1}{3} x \)[/tex] is the inverse of [tex]\( f(x) = 3x \)[/tex], we need to check whether applying both functions in succession results in the original input [tex]\( x \)[/tex]. This means we need to see if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex].
Let's break this down step-by-step for each expression:
1. [tex]\( 3x\left(\frac{x}{3}\right) \)[/tex]:
Substitute [tex]\( g(x) = \frac{1}{3} x \)[/tex] into [tex]\( f(x) = 3x \)[/tex]:
[tex]\[
f(g(x)) = f\left(\frac{1}{3} x\right) = 3 \left(\frac{1}{3} x\right) = x.
\][/tex]
This is not the expression verified here but it sets the understanding.
2. [tex]\( \left(\frac{1}{3} x\right)(3x) \)[/tex]:
Substitute [tex]\( f(x) = 3x \)[/tex] into [tex]\( g(x) = \frac{1}{3} x \)[/tex]:
[tex]\[
g(f(x)) = g(3x) = \frac{1}{3} (3x) = x.
\][/tex]
This confirmation looks reasonable but we move to next.
3. [tex]\( \frac{1}{3}(3 x) \)[/tex]:
Apply [tex]\( f(x) = 3x \)[/tex] first, and then [tex]\( \frac{1}{3} \)[/tex] to the result:
[tex]\[
g(f(x)) = g(3x) = \frac{1}{3}(3 x) = x.
\][/tex]
[tex]\[
f(g(x)) = f\left(\frac{1}{3} x\right) = 3\left(\frac{1}{3} x\right) = x.
\][/tex]
This verifies also correct.
4. [tex]\( \frac{1}{3}\left(\frac{1}{3} x\right) \)[/tex]:
Apply [tex]\( g(x) = \frac{1}{3} x \)[/tex] first, and then apply [tex]\( \frac{1}{3} \)[/tex] to [tex]\( \frac{1}{3} x \)[/tex]:
[tex]\[
g(g(x)) = g\left(\frac{1}{3} x\right) = \frac{1}{3} \left(\frac{1}{3} x\right) = \frac{1}{9} x.
\][/tex]
This does not simplify to [tex]\( x \)[/tex].
Thus, the final step to choose will reveal [tex]\( \frac{1}{3}\left(3 x\right) \)[/tex] simplifies to [tex]\[ x \][/tex] making it the correct simplifier and
the applied expression:
[tex]\[ \boxed{\frac{1}{3}(3 x)} \][/tex]
which confirms [tex]\( g(x) \)[/tex] is indeed the inverse.
