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Let [tex]\( f(x) = -3x - 8 \)[/tex] and [tex]\( g(x) = 5x^2 + 3 \)[/tex].

Find and simplify [tex]\( (f \circ g)(x) \)[/tex].


Sagot :

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]
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