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Given the values of the linear functions [tex]f(x)[/tex] and [tex]g(x)[/tex] in the tables, where is [tex](f-g)(x)[/tex] positive?

\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -10 & -7 & -4 & -1 & 2 \\
\hline
[tex]$f(x)$[/tex] & -3 & 6 & 15 & 24 & 33 \\
\hline
[tex]$g(x)$[/tex] & 9 & 6 & 3 & 0 & -3 \\
\hline
\end{tabular}


Sagot :

Let's solve the problem step-by-step.

1. List the given values from the tables:
- For [tex]\( x \)[/tex] values: [tex]\([-10, -7, -4, -1, 2]\)[/tex]
- For [tex]\( f(x) \)[/tex] values: [tex]\([-3, 6, 15, 24, 33]\)[/tex]
- For [tex]\( g(x) \)[/tex] values: [tex]\([9, 6, 3, 0, -3]\)[/tex]

2. Calculate [tex]\((f-g)(x)\)[/tex] for each corresponding [tex]\( x \)[/tex] value:

- When [tex]\( x = -10 \)[/tex]:
[tex]\[ (f-g)(-10) = f(-10) - g(-10) = -3 - 9 = -12 \][/tex]
- When [tex]\( x = -7 \)[/tex]:
[tex]\[ (f-g)(-7) = f(-7) - g(-7) = 6 - 6 = 0 \][/tex]
- When [tex]\( x = -4 \)[/tex]:
[tex]\[ (f-g)(-4) = f(-4) - g(-4) = 15 - 3 = 12 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ (f-g)(-1) = f(-1) - g(-1) = 24 - 0 = 24 \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ (f-g)(2) = f(2) - g(2) = 33 - (-3) = 33 + 3 = 36 \][/tex]

Collect these results:
[tex]\[ (f - g)(x) = [-12, 0, 12, 24, 36] \][/tex]

3. Identify where [tex]\((f - g)(x)\)[/tex] is positive:

- [tex]\((f - g)(-10) = -12\)[/tex] (not positive)
- [tex]\((f - g)(-7) = 0\)[/tex] (not positive)
- [tex]\((f - g)(-4) = 12\)[/tex] (positive)
- [tex]\((f - g)(-1) = 24\)[/tex] (positive)
- [tex]\((f - g)(2) = 36\)[/tex] (positive)

4. List the [tex]\( x \)[/tex] values where [tex]\((f - g)(x)\)[/tex] is positive:
[tex]\[ x = -4, -1, 2 \][/tex]

Therefore, [tex]\((f - g)(x)\)[/tex] is positive at [tex]\( x = -4, -1, \)[/tex] and [tex]\( 2 \)[/tex].
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