To evaluate the function composition [tex]\((f \circ g)(x)\)[/tex] where [tex]\(f(x) = 3x^2\)[/tex] and [tex]\(g(x) = 2x - 3\)[/tex] at [tex]\(x = -1\)[/tex], we need to follow these steps:
1. First, calculate [tex]\(g(x)\)[/tex] at [tex]\(x = -1\)[/tex]:
[tex]\[
g(-1) = 2(-1) - 3
\][/tex]
[tex]\[
g(-1) = -2 - 3
\][/tex]
[tex]\[
g(-1) = -5
\][/tex]
2. Next, use the result of [tex]\(g(-1)\)[/tex] to find [tex]\(f(g(x))\)[/tex]. Specifically, we need to evaluate [tex]\(f\)[/tex] at [tex]\(g(-1)\)[/tex]:
We have found that [tex]\(g(-1) = -5\)[/tex]. Now, calculate [tex]\(f(-5)\)[/tex]:
[tex]\[
f(-5) = 3(-5)^2
\][/tex]
[tex]\[
f(-5) = 3(25)
\][/tex]
[tex]\[
f(-5) = 75
\][/tex]
Therefore, the composition [tex]\((f \circ g)(-1)\)[/tex] equals [tex]\(75\)[/tex]. The two values of interest are:
[tex]\[
g(-1) = -5
\][/tex]
[tex]\[
(f \circ g)(-1) = 75
\][/tex]
Thus, the final result is:
[tex]\[
g(-1) = -5 \quad \text{and} \quad (f \circ g)(-1) = 75
\][/tex]
