Let's break down the problem step by step:
### Given Functions
- [tex]\( r(x) = 2x^2 \)[/tex]
- [tex]\( s(x) = x - 5 \)[/tex]
### 1. Expression for [tex]\((s + r)(x)\)[/tex]
To find [tex]\((s + r)(x)\)[/tex], we need to add the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
(s + r)(x) = s(x) + r(x)
\][/tex]
Substitute the expressions for [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
(s + r)(x) = (x - 5) + (2x^2)
\][/tex]
Simplify the expression:
[tex]\[
(s + r)(x) = 2x^2 + (x - 5)
\][/tex]
So, the expression for [tex]\((s + r)(x)\)[/tex] is:
[tex]\[
(s + r)(x) = 2x^2 + x - 5
\][/tex]
### 2. Expression for [tex]\((s \cdot r)(x)\)[/tex]
To find [tex]\((s \cdot r)(x)\)[/tex], we need to multiply the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
(s \cdot r)(x) = s(x) \cdot r(x)
\][/tex]
Substitute the expressions for [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
(s \cdot r)(x) = (x - 5) \cdot (2x^2)
\][/tex]
Simplify the expression:
[tex]\[
(s \cdot r)(x) = 2x^2 \cdot (x - 5) = 2x^3 - 10x^2
\][/tex]
So, the expression for [tex]\((s \cdot r)(x)\)[/tex] is:
[tex]\[
(s \cdot r)(x) = 2x^3 - 10x^2
\][/tex]
### 3. Evaluate [tex]\((s - r)(2)\)[/tex]
To find [tex]\((s - r)(2)\)[/tex], we need to subtract the value of [tex]\(r(2)\)[/tex] from [tex]\(s(2)\)[/tex]:
[tex]\[
(s - r)(2) = s(2) - r(2)
\][/tex]
First, calculate [tex]\(s(2)\)[/tex]:
[tex]\[
s(2) = 2 - 5 = -3
\][/tex]
Next, calculate [tex]\(r(2)\)[/tex]:
[tex]\[
r(2) = 2 \cdot 2^2 = 2 \cdot 4 = 8
\][/tex]
Now, subtract [tex]\(r(2)\)[/tex] from [tex]\(s(2)\)[/tex]:
[tex]\[
(s - r)(2) = -3 - 8 = -11
\][/tex]
So, the result for [tex]\((s - r)(2)\)[/tex] is:
[tex]\[
(s - r)(2) = -11
\][/tex]
### Final Results
[tex]\[
\begin{array}{r}
(s+r)(x) = 2x^2 + x - 5 \\
(s \cdot r)(x) = 2x^3 - 10x^2 \\
(s-r)(2) = -11
\end{array}
\][/tex]
