Sure! Let's go through each part of the problem step-by-step.
1. Finding [tex]\((s - t)(x)\)[/tex]:
[tex]\[
s(x) = 4x - 5
\][/tex]
[tex]\[
t(x) = x - 4
\][/tex]
Now, we calculate [tex]\((s - t)(x)\)[/tex]:
[tex]\[
(s - t)(x) = s(x) - t(x) = (4x - 5) - (x - 4)
\][/tex]
Simplify the expression:
[tex]\[
(s - t)(x) = 4x - 5 - x + 4 = 3x - 1
\][/tex]
Therefore,
[tex]\[
(s - t)(x) = 3x - 1
\][/tex]
2. Finding [tex]\((s \cdot t)(x)\)[/tex]:
[tex]\[
s(x) = 4x - 5
\][/tex]
[tex]\[
t(x) = x - 4
\][/tex]
Now, we calculate [tex]\((s \cdot t)(x)\)[/tex]:
[tex]\[
(s \cdot t)(x) = s(x) \cdot t(x) = (4x - 5) \cdot (x - 4)
\][/tex]
Expand the expression:
[tex]\[
(s \cdot t)(x) = 4x(x - 4) - 5(x - 4)
\][/tex]
[tex]\[
(s \cdot t)(x) = 4x^2 - 16x - 5x + 20
\][/tex]
Simplify the expression:
[tex]\[
(s \cdot t)(x) = 4x^2 - 21x + 20
\][/tex]
Therefore,
[tex]\[
(s \cdot t)(x) = 4x^2 - 21x + 20
\][/tex]
3. Evaluating [tex]\((s + t)(-1)\)[/tex]:
[tex]\[
s(x) = 4x - 5
\][/tex]
[tex]\[
t(x) = x - 4
\][/tex]
First, we need to find [tex]\(s(-1)\)[/tex] and [tex]\(t(-1)\)[/tex]:
[tex]\[
s(-1) = 4(-1) - 5 = -4 - 5 = -9
\][/tex]
[tex]\[
t(-1) = -1 - 4 = -5
\][/tex]
Now, we calculate [tex]\((s + t)(-1)\)[/tex]:
[tex]\[
(s + t)(-1) = s(-1) + t(-1) = -9 + (-5) = -14
\][/tex]
Therefore,
[tex]\[
(s + t)(-1) = -14
\][/tex]
To summarize:
[tex]\[
\begin{array}{l}
(s - t)(x) = 3x - 1 \\
(s \cdot t)(x) = 4x^2 - 21x + 20 \\
(s + t)(-1) = -14
\end{array}
\][/tex]
