Let's solve each part step-by-step using the given values in the table:
[tex]\[
\begin{array}{|ccc|}
\hline x & f(x) & g(x) \\
1 & -3 & 2 \\
2 & 3 & 4 \\
3 & 1 & -4 \\
4 & -4 & -1 \\
5 & 2 & 5 \\
\hline
\end{array}
\][/tex]
### a) [tex]\((f - g)(4)\)[/tex]
To find [tex]\((f - g)(4)\)[/tex], we need to subtract [tex]\(g(4)\)[/tex] from [tex]\(f(4)\)[/tex].
From the table:
[tex]\[ f(4) = -4 \][/tex]
[tex]\[ g(4) = -1 \][/tex]
So,
[tex]\[
(f - g)(4) = f(4) - g(4) = -4 - (-1) = -4 + 1 = -3
\][/tex]
Therefore, [tex]\((f - g)(4) = -3\)[/tex].
### b) [tex]\((f + g)(4) - (g - f)(5)\)[/tex]
First, we need to find [tex]\((f + g)(4)\)[/tex] and [tex]\((g - f)(5)\)[/tex].
From the table:
[tex]\[ f(4) = -4 \][/tex]
[tex]\[ g(4) = -1 \][/tex]
Then,
[tex]\[
(f + g)(4) = f(4) + g(4) = -4 + (-1) = -4 - 1 = -5
\][/tex]
Next, from the table:
[tex]\[ f(5) = 2 \][/tex]
[tex]\[ g(5) = 5 \][/tex]
Then,
[tex]\[
(g - f)(5) = g(5) - f(5) = 5 - 2 = 3
\][/tex]
Now, we combine these results:
[tex]\[
(f + g)(4) - (g - f)(5) = -5 - 3 = -8
\][/tex]
### c) [tex]\(\left(\frac{f}{g}\right)(4)\)[/tex]
To find [tex]\(\left(\frac{f}{g}\right)(4)\)[/tex], we need to divide [tex]\(f(4)\)[/tex] by [tex]\(g(4)\)[/tex].
From the table:
[tex]\[ f(4) = -4 \][/tex]
[tex]\[ g(4) = -1 \][/tex]
So,
[tex]\[
\left(\frac{f}{g}\right)(4) = \frac{f(4)}{g(4)} = \frac{-4}{-1} = 4.0
\][/tex]
Therefore, the solutions are:
a) [tex]\( (f - g)(4) = -3 \)[/tex]
b) [tex]\( (f + g)(4) - (g - f)(5) = -8 \)[/tex]
c) [tex]\( \left(\frac{f}{g}\right)(4) = 4.0 \)[/tex]
