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

[tex]\[
\begin{array}{l}
f(x) = 3x \\
g(x) = 2x + 6
\end{array}
\][/tex]

Write the expressions for [tex]\( (g - f)(x) \)[/tex] and [tex]\( (g \cdot f)(x) \)[/tex] and evaluate [tex]\( (g + f)(3) \)[/tex].

[tex]\[
\begin{array}{c}
(g - f)(x) = \\
(g \cdot f)(x) = \\
(g + f)(3) =
\end{array}
\][/tex]

[tex]\[ \square \][/tex]
[tex]\[ \square \][/tex]
[tex]\[ \square \][/tex]



Answer :

GinnyAnswer
Let's solve this step-by-step.

1. Find the expression for [tex]\((g - f)(x)\)[/tex]:

[tex]\(f(x) = 3x\)[/tex]
[tex]\(g(x) = 2x + 6\)[/tex]

[tex]\[ (g - f)(x) = g(x) - f(x) = (2x + 6) - 3x = 2x + 6 - 3x = -x + 6 \][/tex]

So, [tex]\((g - f)(x) = -x + 6\)[/tex].

2. Find the expression for [tex]\((g \cdot f)(x)\)[/tex]:

[tex]\[ (g \cdot f)(x) = g(x) \cdot f(x) = (2x + 6) \cdot 3x = 6x^2 + 18x \][/tex]

So, [tex]\((g \cdot f)(x) = 6x^2 + 18x\)[/tex].

3. Evaluate [tex]\((g + f)(3)\)[/tex]:

[tex]\[ g(x) = 2x + 6 \][/tex]
[tex]\[ f(x) = 3x \][/tex]

[tex]\[ (g + f)(x) = g(x) + f(x) = (2x + 6) + 3x = 2x + 6 + 3x = 5x + 6 \][/tex]

To evaluate [tex]\((g + f)(3)\)[/tex]:

[tex]\[ (g + f)(3) = 5(3) + 6 = 15 + 6 = 21 \][/tex]

So, [tex]\((g + f)(3) = 21\)[/tex].

Therefore, the results are:
[tex]\[ \begin{aligned} (g - f)(x) &= -x + 6 \\ (g \cdot f)(x) &= 6x^2 + 18x \\ (g + f)(3) &= 21 \end{aligned} \][/tex]

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