To solve the problem, let's first define the given functions:
[tex]\[ r(x) = 2x \][/tex]
[tex]\[ s(x) = 4x^2 \][/tex]
### 1. Expression for [tex]\((r - s)(x)\)[/tex]:
[tex]\[ (r - s)(x) = r(x) - s(x) \][/tex]
[tex]\[ (r - s)(x) = 2x - 4x^2 \][/tex]
So, the expression for [tex]\((r - s)(x)\)[/tex] is:
[tex]\[ (r - s)(x) = 2x - 4x^2 \][/tex]
### 2. Expression for [tex]\((r \cdot s)(x)\)[/tex]:
[tex]\[ (r \cdot s)(x) = r(x) \cdot s(x) \][/tex]
[tex]\[ (r \cdot s)(x) = (2x) \cdot (4x^2) \][/tex]
[tex]\[ (r \cdot s)(x) = 8x^3 \][/tex]
So, the expression for [tex]\((r \cdot s)(x)\)[/tex] is:
[tex]\[ (r \cdot s)(x) = 8x^3 \][/tex]
### 3. Evaluation of [tex]\((r + s)(-2)\)[/tex]:
[tex]\[ (r + s)(x) = r(x) + s(x) \][/tex]
First, let's evaluate [tex]\(r(-2)\)[/tex]:
[tex]\[ r(-2) = 2 \cdot (-2) = -4 \][/tex]
Now, let's evaluate [tex]\(s(-2)\)[/tex]:
[tex]\[ s(-2) = 4 \cdot (-2)^2 = 4 \cdot 4 = 16 \][/tex]
Now, let's find [tex]\((r + s)(-2)\)[/tex]:
[tex]\[ (r + s)(-2) = r(-2) + s(-2) \][/tex]
[tex]\[ (r + s)(-2) = -4 + 16 = 12 \][/tex]
So, [tex]\((r + s)(-2) = 12\)[/tex].
Putting it all together:
[tex]\[
\begin{aligned}
(r - s)(x) &= 2x - 4x^2 \\
(r \cdot s)(x) &= 8x^3 \\
(r + s)(-2) &= 12
\end{aligned}
\][/tex]
