Certainly! To find the composite function [tex]\((F \circ g)(x)\)[/tex], we need to evaluate [tex]\(F\)[/tex] at [tex]\(g(x)\)[/tex]. That is, we need to plug [tex]\(g(x)\)[/tex] into [tex]\(F(x)\)[/tex].
1. Define the functions:
- Given:
[tex]\[
F(x) = 5x - 1
\][/tex]
[tex]\[
g(x) = x^3 - 4x + 2
\][/tex]
2. Find [tex]\(g(x)\)[/tex] for specific values of [tex]\(x\)[/tex]:
- For [tex]\(x = 0\)[/tex]:
[tex]\[
g(0) = (0)^3 - 4(0) + 2 = 2
\][/tex]
- For [tex]\(x = 1\)[/tex]:
[tex]\[
g(1) = (1)^3 - 4(1) + 2 = 1 - 4 + 2 = -1
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
g(2) = (2)^3 - 4(2) + 2 = 8 - 8 + 2 = 2
\][/tex]
- For [tex]\(x = 3\)[/tex]:
[tex]\[
g(3) = (3)^3 - 4(3) + 2 = 27 - 12 + 2 = 17
\][/tex]
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(F(x)\)[/tex]:
- For [tex]\(x = 0\)[/tex]:
[tex]\[
F(g(0)) = F(2) = 5(2) - 1 = 10 - 1 = 9
\][/tex]
- For [tex]\(x = 1\)[/tex]:
[tex]\[
F(g(1)) = F(-1) = 5(-1) - 1 = -5 - 1 = -6
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
F(g(2)) = F(2) = 5(2) - 1 = 10 - 1 = 9
\][/tex]
- For [tex]\(x = 3\)[/tex]:
[tex]\[
F(g(3)) = F(17) = 5(17) - 1 = 85 - 1 = 84
\][/tex]
Therefore, the values of the composite function [tex]\((F \circ g)(x)\)[/tex] for [tex]\(x = 0, 1, 2, 3\)[/tex] are:
[tex]\[
\begin{aligned}
(F \circ g)(0) &= 9 \\
(F \circ g)(1) &= -6 \\
(F \circ g)(2) &= 9 \\
(F \circ g)(3) &= 84 \\
\end{aligned}
\][/tex]
Thus, [tex]\((F \circ g)(x)\)[/tex] evaluated at [tex]\(x = 0, 1, 2, 3\)[/tex] yields the results [tex]\(9, -6, 9, 84\)[/tex] respectively.
