Question

the functions represent the amounts of water released from two reservoirs over x weeks: reservoir a \\( f(x) = x^2 - 7x + 5 \\) reservoir b \\( g(x) = 3x^2 - 6x + 2 \\) the function \\( h(x) = f(x) - g(x) \\) represents the difference in the amounts of water released. determine which statements about \\( h(x) \\) and about the reservoirs are true. choose two correct answers. \\( \square \\) reservoir a releases less water than reservoir b over 1 week. \\( \square \\) \\( h(x) = -2x^2 - x + 3 \\) \\( \square \\) reservoir a releases more water than reservoir b over 1 week. \\( \square \\) reservoir a releases the same amount of water as reservoir b over 1 week. \\( \square \\) \\( h(x) = -2x^2 - 13x + 6 \\)

Explanation:

Step1: Calculate \( h(x) \)

Given \( f(x)=x^{2}-7x + 6 \) and \( g(x)=3x^{2}-6x + 2 \), then \( h(x)=f(x)-g(x) \).
\[

$$\begin{align*} h(x)&=(x^{2}-7x + 6)-(3x^{2}-6x + 2)\\ &=x^{2}-7x + 6-3x^{2}+6x - 2\\ &=(x^{2}-3x^{2})+(-7x + 6x)+(6 - 2)\\ &=-2x^{2}-x + 3 \end{align*}$$

\]

Step2: Evaluate \( f(1) \) and \( g(1) \)

For \( f(1) \):
\[
f(1)=1^{2}-7\times1 + 6=1-7 + 6=0
\]
For \( g(1) \):
\[
g(1)=3\times1^{2}-6\times1 + 2=3-6 + 2=-1
\]
Since \( f(1) = 0\) and \( g(1)=- 1\), and \(0>-1\), so Reservoir A releases more water than Reservoir B over 1 week. Also, from Step 1, \( h(x)=-2x^{2}-x + 3 \) is correct.

Answer:

1. \( h(x)=-2x^{2}-x + 3 \)
2. Reservoir A releases more water than Reservoir B over 1 week.