To find and simplify the composed function [tex]\((f \circ g)(x)\)[/tex], we need to evaluate [tex]\(f(g(x))\)[/tex]. Here's the step-by-step process:
1. Identify the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = -3x - 8
\][/tex]
[tex]\[
g(x) = 5x^2 + 3
\][/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex] to get the composed function [tex]\(f(g(x))\)[/tex]:
We aim to find [tex]\(f(g(x))\)[/tex], which means we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(5x^2 + 3)
\][/tex]
3. Apply [tex]\(f(x)\)[/tex] to [tex]\(g(x)\)[/tex]:
Since [tex]\(f(x) = -3x - 8\)[/tex], replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex]:
[tex]\[
f(5x^2 + 3) = -3(5x^2 + 3) - 8
\][/tex]
4. Simplify the expression:
First, distribute the [tex]\(-3\)[/tex]:
[tex]\[
-3(5x^2 + 3) = -3 \cdot 5x^2 - 3 \cdot 3 = -15x^2 - 9
\][/tex]
Then, subtract 8 from the result:
[tex]\[
-15x^2 - 9 - 8 = -15x^2 - 17
\][/tex]
Therefore, the composed function [tex]\((f \circ g)(x)\)[/tex] simplifies to:
[tex]\[
(f \circ g)(x) = -15x^2 - 17
\][/tex]
