Let's examine the given table and the expression that represents \( h(x) \) step-by-step.
We are provided with the following table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & f(x) = 2x + 1 & g(x) = -x - 3 & h(x) \\
\hline
-3 & -5 & 0 & 5 \\
\hline
2 & 5 & -5 & -10 \\
\hline
4 & 9 & -7 & -16 \\
\hline
\end{array}
\][/tex]
We need to determine which of the following expressions matches \( h(x) \):
1. \( (f + g)(x) \)
2. \( (g + f)(x) \)
3. \( (f - g)(x) \)
4. \( (g - f)(x) \)
To do this, we will evaluate each given expression step by step for the provided \( x \) values: \( -3, 2, \) and \( 4 \).
### Step-by-Step Calculation
1. \( (f + g)(x) \):
\( (f + g)(x) = (2x + 1) + (-x - 3) \)
[tex]\[
(f + g)(x) = 2x + 1 - x - 3 = x - 2
\][/tex]
Let's calculate \( (f + g)(x) \) for each \( x \):
- For \( x = -3 \): \( (f + g)(-3) = -3 - 2 = -5 \neq 5 \)
- For \( x = 2 \): \( (f + g)(2) = 2 - 2 = 0 \neq -10 \)
- For \( x = 4 \): \( (f + g)(4) = 4 - 2 = 2 \neq -16 \)
2. \( (g + f)(x) \):
This is just another notation of \( (f + g)(x) \). The results would be the same as above:
[tex]\[
(g + f)(x) = (f + g)(x) = x - 2
\][/tex]
3. \( (f - g)(x) \):
\( (f - g)(x) = (2x + 1) - (-x - 3) \)
[tex]\[
(f - g)(x) = 2x + 1 + x + 3 = 3x + 4
\][/tex]
Let's calculate \( (f - g)(x) \) for each \( x \):
- For \( x = -3 \): \( (f - g)(-3) = 3(-3) + 4 = -9 + 4 = -5 \neq 5 \)
- For \( x = 2 \): \( (f - g)(2) = 3(2) + 4 = 6 + 4 = 10 \neq -10 \)
- For \( x = 4 \): \( (f - g)(4) = 3(4) + 4 = 12 + 4 = 16 \neq -16 \)
4. \( (g - f)(x) \):
\( (g - f)(x) = (-x - 3) - (2x + 1) \)
[tex]\[
(g - f)(x) = -x - 3 - 2x - 1 = -3x - 4
\][/tex]
Let's calculate \( (g - f)(x) \) for each \( x \):
- For \( x = -3 \): \( (g - f)(-3) = -3(-3) - 4 = 9 - 4 = 5 = 5 \)
- For \( x = 2 \): \( (g - f)(2) = -3(2) - 4 = -6 - 4 = -10 = -10 \)
- For \( x = 4 \): \( (g - f)(4) = -3(4) - 4 = -12 - 4 = -16 = -16 \)
### Conclusion
The expression \( (g - f)(x) \) matches \( h(x) \) for all given \( x \) values.
Thus, the expression that represents [tex]\( h(x) \)[/tex] is [tex]\( (g - f)(x) \)[/tex].
