To find [tex]\( (f \circ g)(x) \)[/tex], which denotes the composition of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], we need to evaluate [tex]\( f(g(x)) \)[/tex].
Given:
[tex]\[ f(x) = 2x - 7 \][/tex]
[tex]\[ g(x) = 4x^2 + 2x + 6 \][/tex]
Let's break it down step-by-step:
1. Evaluate [tex]\( g(x) \)[/tex] first:
We need to determine [tex]\( g(x) \)[/tex] for a specific value of [tex]\( x \)[/tex]. We'll use [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 4(1)^2 + 2(1) + 6
\][/tex]
Calculate the value:
[tex]\[
g(1) = 4 \cdot 1 + 2 \cdot 1 + 6
\][/tex]
[tex]\[
g(1) = 4 + 2 + 6
\][/tex]
[tex]\[
g(1) = 12
\][/tex]
Therefore, [tex]\( g(1) = 12 \)[/tex].
2. Now evaluate [tex]\( f(g(x)) \)[/tex]:
We need to find [tex]\( f \)[/tex] at the value [tex]\( g(1) \)[/tex], which we found to be 12. So, evaluate [tex]\( f(12) \)[/tex]:
[tex]\[
f(12) = 2(12) - 7
\][/tex]
Calculate the value:
[tex]\[
f(12) = 24 - 7
\][/tex]
[tex]\[
f(12) = 17
\][/tex]
Hence, [tex]\( f(g(1)) = 17 \)[/tex].
From the steps above, we have determined:
- [tex]\( g(1) = 12 \)[/tex]
- [tex]\( (f \circ g)(1) = 17 \)[/tex]
Therefore, the result of the composition is:
[tex]\[ (f \circ g)(x) = 17 \quad \text{when} \quad x = 1 \][/tex]
