Answered

If [tex] f(x) = 5x - 25 [/tex] and [tex] g(x) = \frac{1}{5}x + 5 [/tex], which expression could be used to verify [tex] g(x) [/tex] is the inverse of [tex] f(x) [/tex]?

A. [tex] \frac{1}{5}\left(\frac{1}{5}x + 5\right) + 5 [/tex]

B. [tex] \frac{1}{5}(5x - 25) + 5 [/tex]

C. [tex] \frac{1}{\left(\frac{1}{5}x + 5\right)} [/tex]

D. [tex] 5\left(\frac{1}{5}x + 5\right) + 5 [/tex]



Answer :

GinnyAnswer
To verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to check if applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex] yields the original input, [tex]\( x \)[/tex]. In mathematical terms, we need to find out if [tex]\( g(f(x)) = x \)[/tex].

Given:
[tex]\[ f(x) = 5x - 25 \][/tex]
[tex]\[ g(x) = \frac{1}{5} x + 5 \][/tex]

Let's determine [tex]\( g(f(x)) \)[/tex]:

1. First, apply [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 5x - 25 \][/tex]

2. Now apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]

Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 \][/tex]

Simplify the expression step-by-step:
[tex]\[ g(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 + 5 \][/tex]
[tex]\[ g(5x - 25) = x - 5 + 5 \][/tex]
[tex]\[ g(5x - 25) = x \][/tex]

We have shown that [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].

The correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]

So, the correct answer is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]

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