Answered

If [tex]\( f(x) = 3x \)[/tex] and [tex]\( g(x) = \frac{1}{3}x \)[/tex], which expression could be used to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex]?

A. [tex]\( 3x \left( \frac{\pi}{3} \right) \)[/tex]
B. [tex]\( \left( \frac{1}{3}x \right)(3x) \)[/tex]
C. [tex]\( \frac{1}{3}(3x) \)[/tex]
D. [tex]\( \frac{1}{3} \left( \frac{1}{3}x \right) \)[/tex]



Answer :

GinnyAnswer
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]

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