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Suppose that the functions [tex]\( r \)[/tex] and [tex]\( s \)[/tex] are defined for all real numbers [tex]\( x \)[/tex] as follows:
[tex]\[
\begin{array}{l}
r(x) = 3x - 2 \\
s(x) = 6x
\end{array}
\][/tex]

Write the expressions for [tex]\((r \cdot s)(x)\)[/tex] and [tex]\((r - s)(x)\)[/tex], and evaluate [tex]\((r + s)(3)\)[/tex].

[tex]\[
\begin{array}{l}
(r \cdot s)(x) = \square \\
(r - s)(x) = \square \\
(r + s)(3) = \square
\end{array}
\][/tex]

[tex]\(\square\)[/tex]

Sagot :

GinnyAnsweridn
Let's start by writing the given functions and follow the steps to derive the required expressions and evaluate the function at a given point.

Given functions:
[tex]\[ r(x) = 3x - 2 \][/tex]
[tex]\[ s(x) = 6x \][/tex]

1. Expression for [tex]\((r \cdot s)(x)\)[/tex]:
[tex]\[ (r \cdot s)(x) = r(x) \cdot s(x) \][/tex]
Substituting the given functions:
[tex]\[ r(x) = 3x - 2 \quad \text{and} \quad s(x) = 6x \][/tex]
Therefore,
[tex]\[ (r \cdot s)(x) = (3x - 2) \cdot (6x) \][/tex]
Multiplying the expressions:
[tex]\[ (r \cdot s)(x) = 18x^2 - 12x \][/tex]

2. Expression for [tex]\((r - s)(x)\)[/tex]:
[tex]\[ (r - s)(x) = r(x) - s(x) \][/tex]
Substituting the given functions:
[tex]\[ r(x) = 3x - 2 \quad \text{and} \quad s(x) = 6x \][/tex]
Therefore,
[tex]\[ (r - s)(x) = (3x - 2) - (6x) \][/tex]
Simplifying the expression:
[tex]\[ (r - s)(x) = 3x - 2 - 6x \][/tex]
[tex]\[ (r - s)(x) = -3x - 2 \][/tex]

3. Evaluating [tex]\((r + s)(3)\)[/tex]:
[tex]\[ (r + s)(x) = r(x) + s(x) \][/tex]
Substituting the given functions:
[tex]\[ r(x) = 3x - 2 \quad \text{and} \quad s(x) = 6x \][/tex]
Therefore,
[tex]\[ (r + s)(x) = (3x - 2) + (6x) \][/tex]
Simplifying the expression:
[tex]\[ (r + s)(x) = 3x - 2 + 6x \][/tex]
[tex]\[ (r + s)(x) = 9x - 2 \][/tex]
Evaluating this at [tex]\(x = 3\)[/tex]:
[tex]\[ (r + s)(3) = 9(3) - 2 \][/tex]
[tex]\[ (r + s)(3) = 27 - 2 \][/tex]
[tex]\[ (r + s)(3) = 25 \][/tex]

Thus, we have:
[tex]\[ (r \cdot s)(x) = 18x^2 - 12x \][/tex]
[tex]\[ (r - s)(x) = -3x - 2 \][/tex]
[tex]\[ (r + s)(3) = 25 \][/tex]
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