Answered

Suppose that the functions [tex]f[/tex] and [tex]g[/tex] are defined for all real numbers [tex]x[/tex] as follows.

[tex]\[
\begin{array}{l}
f(x) = 5x^3 \\
g(x) = x^2
\end{array}
\][/tex]

Write the expressions for [tex](f \cdot g)(x)[/tex] and [tex](f + g)(x)[/tex] and evaluate [tex](f - g)(-2)[/tex].

[tex]\[
\begin{aligned}
(f \cdot g)(x) & = \square \\
(f + g)(x) & = \square \\
(f - g)(-2) & = \square
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
Let's go through each part of the question step-by-step.

1. Expression for [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
Given the functions:
[tex]\[ f(x) = 5x^3 \][/tex]
[tex]\[ g(x) = x^2 \][/tex]
Substitute for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = (5x^3) \cdot (x^2) \][/tex]
[tex]\[ (f \cdot g)(x) = 5x^5 \][/tex]

2. Expression for [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substitute for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f+g)(x) = 5x^3 + x^2 \][/tex]

3. Evaluate [tex]\((f-g)(-2)\)[/tex]:
[tex]\[ (f-g)(x) = f(x) - g(x) \][/tex]
First, determine [tex]\( f(-2) \)[/tex] and [tex]\( g(-2) \)[/tex]:
[tex]\[ f(-2) = 5(-2)^3 = 5(-8) = -40 \][/tex]
[tex]\[ g(-2) = (-2)^2 = 4 \][/tex]
Then, evaluate [tex]\((f-g)(-2)\)[/tex]:
[tex]\[ (f-g)(-2) = f(-2) - g(-2) = -40 - 4 = -44 \][/tex]

Hence, the results are:
[tex]\[ (f \cdot g)(x) = 5x^5 \][/tex]
[tex]\[ (f+g)(x) = 5x^3 + x^2 \][/tex]
[tex]\[ (f-g)(-2) = -44 \][/tex]

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