If [tex]h(x) = x - 7[/tex] and [tex]g(x) = x^2[/tex], which expression is equivalent to [tex](g \circ h)(5)[/tex]?

A. [tex](5-7)^2[/tex]
B. [tex]5^2 - 7[/tex]
C. [tex]5^2 (5-7)[/tex]
D. [tex](5-7) x^2[/tex]



Answer :

Sure, let's break it down step-by-step. We are given [tex]\( h(x) = x - 7 \)[/tex] and [tex]\( g(x) = x^2 \)[/tex].
We need to find the equivalent expression for [tex]\( (g \circ h)(5) \)[/tex].

The notation [tex]\( (g \circ h)(5) \)[/tex] represents the composition of the functions [tex]\( g \)[/tex] and [tex]\( h \)[/tex], applied to the input 5. We perform this in two steps:

1. First, apply the function [tex]\( h(x) \)[/tex] to the input 5:
[tex]\[ h(5) = 5 - 7 \][/tex]
Simplifying this, we obtain:
[tex]\[ h(5) = -2 \][/tex]

2. Next, take the result from the first step and apply the function [tex]\( g(x) \)[/tex] to it:
[tex]\[ g(h(5)) = g(-2) \][/tex]
Since [tex]\( g(x) = x^2 \)[/tex], we have:
[tex]\[ g(-2) = (-2)^2 \][/tex]

3. Simplify the expression [tex]\( (-2)^2 \)[/tex]:
[tex]\[ (-2)^2 = 4 \][/tex]

So, the value of [tex]\( (g \circ h)(5) \)[/tex] is 4. Therefore, the equivalent expression is:

[tex]\[ (5 - 7)^2 \][/tex]