Let's go through each part of the problem step-by-step.
### Part 1: 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]\[ s(x) = x - 1 \][/tex]
[tex]\[ r(x) = 3x + 1 \][/tex]
We calculate the product:
[tex]\[
(s \cdot r)(x) = s(x) \cdot r(x) = (x - 1) \cdot (3x + 1)
\][/tex]
Distribute the terms:
[tex]\[
(x - 1)(3x + 1) = x \cdot 3x + x \cdot 1 - 1 \cdot 3x - 1 \cdot 1
\][/tex]
[tex]\[
= 3x^2 + x - 3x - 1
\][/tex]
[tex]\[
= 3x^2 - 2x - 1
\][/tex]
Thus, we have:
[tex]\[
(s \cdot r)(x) = 3x^2 - 2x - 1
\][/tex]
### Part 2: Expression for [tex]\((s - r)(x)\)[/tex]
To find [tex]\((s - r)(x)\)[/tex], we need to subtract the function [tex]\(r(x)\)[/tex] from [tex]\(s(x)\)[/tex]:
[tex]\[
s(x) = x - 1
\][/tex]
[tex]\[
r(x) = 3x + 1
\][/tex]
We calculate the difference:
[tex]\[
(s - r)(x) = s(x) - r(x) = (x - 1) - (3x + 1)
\][/tex]
Simplify the expression:
[tex]\[
(x - 1) - (3x + 1) = x - 1 - 3x - 1
\][/tex]
[tex]\[
= x - 3x - 1 - 1
\][/tex]
[tex]\[
= -2x - 2
\][/tex]
Thus, we have:
[tex]\[
(s - r)(x) = -2x - 2
\][/tex]
### Part 3: Evaluating [tex]\((s + r)(1)\)[/tex]
To find [tex]\((s + r)(1)\)[/tex], we need to add the values of [tex]\(s(x)\)[/tex] and [tex]\(r(x)\)[/tex] when [tex]\(x = 1\)[/tex]:
[tex]\[
s(x) = x - 1 \implies s(1) = 1 - 1 = 0
\][/tex]
[tex]\[
r(x) = 3x + 1 \implies r(1) = 3 \cdot 1 + 1 = 4
\][/tex]
We calculate the sum:
[tex]\[
(s + r)(1) = s(1) + r(1) = 0 + 4 = 4
\][/tex]
### Summary
The results are:
[tex]\[
\begin{array}{|l|}
\hline
(s \cdot r)(x) = 3x^2 - 2x - 1 \\
(s - r)(x) = -2x - 2 \\
(s + r)(1) = 4 \\
\hline
\end{array}
\][/tex]
