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Suppose the functions [tex]\( g \)[/tex] and [tex]\( h \)[/tex] are defined for all real numbers [tex]\( x \)[/tex] as follows:

[tex]\[
\begin{array}{l}
g(x) = x + 3 \\
h(x) = 4x^2
\end{array}
\][/tex]

Write the expressions for [tex]\( (g - h)(x) \)[/tex] and [tex]\( (g + h)(x) \)[/tex] and evaluate [tex]\( (g \cdot h)(-1) \)[/tex].

[tex]\[
\begin{array}{l}
(g - h)(x) = \square \\
(g + h)(x) = \square \\
(g \cdot h)(-1) = \square
\end{array}
\][/tex]



Answer :

GinnyAnswer
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]

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