Let's tackle the problems step by step:
8. Finding the Composite Function [tex]\((g \circ f)(x)\)[/tex]:
To find the composite function [tex]\((g \circ f)(x)\)[/tex], we need to substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]. This means:
[tex]\[ (g \circ f)(x) = g(f(x)) \][/tex]
Given the functions:
[tex]\[ f(x) = 5 - 2x^2 + x \][/tex]
[tex]\[ g(x) = \sqrt{x} \][/tex]
We substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ (g \circ f)(x) = g(5 - 2x^2 + x) \][/tex]
Since [tex]\(g(x) = \sqrt{x}\)[/tex], replacing [tex]\(x\)[/tex] with [tex]\(5 - 2x^2 + x\)[/tex] gives us:
[tex]\[ (g \circ f)(x) = \sqrt{5 - 2x^2 + x} \][/tex]
So, the composite function [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = \sqrt{5 - 2x^2 + x} \][/tex]
9. Finding [tex]\((g \circ f)(1)\)[/tex]:
To find [tex]\((g \circ f)(1)\)[/tex], we substitute [tex]\(x = 1\)[/tex] into the composite function [tex]\((g \circ f)(x)\)[/tex]:
[tex]\[ (g \circ f)(1) = \sqrt{5 - 2(1)^2 + 1} \][/tex]
Evaluate the expression inside the square root:
[tex]\[ 5 - 2(1)^2 + 1 = 5 - 2 + 1 = 4 \][/tex]
Now, taking the square root of 4:
[tex]\[ (g \circ f)(1) = \sqrt{4} = 2 \][/tex]
So, [tex]\((g \circ f)(1)\)[/tex] is:
[tex]\[ (g \circ f)(1) = 2 \][/tex]
In summary:
8. The composite function [tex]\((g \circ f)(x)\)[/tex] is [tex]\(\sqrt{5 - 2x^2 + x}\)[/tex].
9. The value of [tex]\((g \circ f)(1)\)[/tex] is [tex]\(2\)[/tex].
