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[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]

Use the table defining [tex]$f$[/tex] and [tex]$g$[/tex] to solve:

(Note: Write "Does not exist" if the value does not exist.)

a) [tex]$(f-g)(4)=$[/tex] [tex]$\square$[/tex]

b) [tex]$(f+g)(4)-(g-f)(5)=$[/tex] [tex]$\square$[/tex]

c) [tex]$\left(\frac{f}{g}\right)(4)=$[/tex] [tex]$\square$[/tex]


Sagot :

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]
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