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 that [tex]\( g(x) = \frac{1}{5}x + 5 \)[/tex] is the inverse of [tex]\( f(x) = 5x - 25 \)[/tex], we need to check if the composition [tex]\( g(f(x)) \)[/tex] simplifies to [tex]\( x \)[/tex]. Let's examine this step-by-step:

1. Define the composition [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Since [tex]\( g(x) = \frac{1}{5}x + 5 \)[/tex], we substitute [tex]\( 5x - 25 \)[/tex] in place of [tex]\( x \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5}(5x - 25) + 5 \][/tex]

2. Simplify the expression [tex]\( \frac{1}{5}(5x - 25) + 5 \)[/tex]:
[tex]\[ \frac{1}{5}(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 = x - 5 \][/tex]
Thus,
[tex]\[ \frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x \][/tex]

So, the simplified expression [tex]\( \frac{1}{5}(5x - 25) + 5 \)[/tex] is indeed [tex]\( x \)[/tex]. This verifies that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].

Therefore, the given expression that verifies [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]

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