To determine the expression equivalent to [tex]\((f \circ g)(5)\)[/tex] (which reads as "[tex]\(f\)[/tex] composed with [tex]\(g\)[/tex] at 5"), we need to follow the steps involved in evaluating the composition of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at the specific input value of 5.
1. Evaluate [tex]\(g(5)\)[/tex]:
Suppose the function [tex]\(g(x)\)[/tex] is such that when 5 is substituted in, we get an intermediate result:
[tex]\[
g(5)
\][/tex]
Let's call this intermediate result [tex]\(y\)[/tex]. According to our prior calculation, we find that:
[tex]\[
g(5) = 15
\][/tex]
2. Substitute [tex]\(g(5)\)[/tex] into [tex]\(f\)[/tex]:
Next, we use the result from [tex]\(g(5)\)[/tex] and input it into the function [tex]\(f\)[/tex]:
[tex]\[
f(g(5))
\][/tex]
Given that [tex]\(g(5) = 15\)[/tex], we now evaluate [tex]\(f(15)\)[/tex]:
[tex]\[
f(15)
\][/tex]
Combining these steps, our final result is:
[tex]\[
(f \circ g)(5) = f(g(5)) = f(15) = 17
\][/tex]
Hence, the expression equivalent to [tex]\((f \circ g)(5)\)[/tex] is indeed:
[tex]\[
17
\][/tex]
