To evaluate [tex]\( (f \circ g)(1) \)[/tex], we need to find the value of [tex]\( f(g(1)) \)[/tex]. Let's break this down step-by-step:
1. Evaluate [tex]\( g(1) \)[/tex]:
Given the function [tex]\( g(x) = 5x^2 - 2 \)[/tex], we substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 5(1)^2 - 2 = 5 \cdot 1 - 2 = 5 - 2 = 3
\][/tex]
2. Evaluate [tex]\( f \)[/tex] at the result of [tex]\( g(1) \)[/tex]:
We have found that [tex]\( g(1) = 3 \)[/tex]. Now, we need to evaluate [tex]\( f \)[/tex] at this value. The function [tex]\( f(x) = 3x + 4 \)[/tex] allows us to do so by substituting [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 3 \cdot 3 + 4 = 9 + 4 = 13
\][/tex]
Thus, [tex]\( (f \circ g)(1) = f(g(1)) = 13 \)[/tex].
Therefore, the evaluated results are [tex]\( g(1) = 3 \)[/tex] and [tex]\( (f \circ g)(1) = 13 \)[/tex].
