Let's determine the expressions [tex]\((g - h)(x)\)[/tex] and [tex]\((g + h)(x)\)[/tex], and evaluate [tex]\((g \cdot h)(-1)\)[/tex] for the given functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex].
1. Expression for [tex]\((g - h)(x)\)[/tex]
Given:
[tex]\[
g(x) = x + 3
\][/tex]
[tex]\[
h(x) = 4x^2
\][/tex]
To find [tex]\((g - h)(x)\)[/tex], we subtract [tex]\(h(x)\)[/tex] from [tex]\(g(x)\)[/tex]:
[tex]\[
(g - h)(x) = g(x) - h(x)
\][/tex]
[tex]\[
(g - h)(x) = (x + 3) - 4x^2
\][/tex]
Therefore:
[tex]\[
(g - h)(x) = -4x^2 + x + 3
\][/tex]
2. Expression for [tex]\((g + h)(x)\)[/tex]
To find [tex]\((g + h)(x)\)[/tex], we add [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex]:
[tex]\[
(g + h)(x) = g(x) + h(x)
\][/tex]
[tex]\[
(g + h)(x) = (x + 3) + 4x^2
\][/tex]
Therefore:
[tex]\[
(g + h)(x) = 4x^2 + x + 3
\][/tex]
3. Evaluate [tex]\((g \cdot h)(-1)\)[/tex]
To find [tex]\((g \cdot h)(-1)\)[/tex], we multiply [tex]\(g(-1)\)[/tex] and [tex]\(h(-1)\)[/tex].
First, calculate [tex]\(g(-1)\)[/tex]:
[tex]\[
g(-1) = (-1) + 3 = 2
\][/tex]
Next, calculate [tex]\(h(-1)\)[/tex]:
[tex]\[
h(-1) = 4(-1)^2 = 4 \cdot 1 = 4
\][/tex]
Now, multiply [tex]\(g(-1)\)[/tex] and [tex]\(h(-1)\)[/tex]:
[tex]\[
(g \cdot h)(-1) = g(-1) \cdot h(-1)
\][/tex]
[tex]\[
(g \cdot h)(-1) = 2 \cdot 4 = 8
\][/tex]
Summarizing the results:
[tex]\[
(g - h)(x) = -4x^2 + x + 3
\][/tex]
[tex]\[
(g + h)(x) = 4x^2 + x + 3
\][/tex]
[tex]\[
(g \cdot h)(-1) = 8
\][/tex]
