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The functions below represent the amounts of water released from two reservoirs over [tex]x[/tex] weeks:

Reservoir A:
[tex]\[ f(x) = x^2 - 7x + 5 \][/tex]

Reservoir B:
[tex]\[ g(x) = 3x^2 - 6x + 2 \][/tex]

The function [tex]h(x) = f(x) - g(x)[/tex] represents the difference in the amounts of water released.

Determine which statements about [tex]h(x)[/tex] and about the reservoirs are true. Check all that apply.

A. [tex]h(x) = -2x^2 - 13x + 3[/tex]
B. [tex]h(x) = -2x^2 - x + 3[/tex]
C. Reservoir A releases less water than Reservoir B over 1 week.
D. Reservoir A releases the same amount of water as Reservoir B over 1 week.
E. Reservoir A releases more water than Reservoir B over 1 week.

Sagot :

GinnyAnswer
Let's analyze the functions given for the released amounts of water from the two reservoirs.

Reservoir A:
[tex]\[ f(x) = x^2 - 7x + 5 \][/tex]

Reservoir B:
[tex]\[ g(x) = 3x^2 - 6x + 2 \][/tex]

The function representing the difference between the amounts of water released by the two reservoirs is:
[tex]\[ h(x) = f(x) - g(x) \][/tex]

First, let's find the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] when [tex]\( x = 1 \)[/tex] week:

For Reservoir A:
[tex]\[ f(1) = 1^2 - 7 \cdot 1 + 5 \][/tex]
[tex]\[ f(1) = 1 - 7 + 5 \][/tex]
[tex]\[ f(1) = -1 \][/tex]

For Reservoir B:
[tex]\[ g(1) = 3 \cdot 1^2 - 6 \cdot 1 + 2 \][/tex]
[tex]\[ g(1) = 3 - 6 + 2 \][/tex]
[tex]\[ g(1) = -1 \][/tex]

Now, let's calculate [tex]\( h(1) \)[/tex]:

[tex]\[ h(1) = f(1) - g(1) \][/tex]
[tex]\[ h(1) = -1 - (-1) \][/tex]
[tex]\[ h(1) = -1 + 1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]

Given the results:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ h(1) = 0 \][/tex]

Let’s evaluate the given statements one-by-one:

1. [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex]:

Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 13(1) + 6 \][/tex]
[tex]\[ h(1) = -2 - 13 + 6 \][/tex]
[tex]\[ h(1) = -9 \neq 0 \][/tex]

The first statement is False.

2. [tex]\( h(x) = -2x^2 - x + 3 \)[/tex]:

Check if this is true for [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = -2(1)^2 - 1(1) + 3 \][/tex]
[tex]\[ h(1) = -2 - 1 + 3 \][/tex]
[tex]\[ h(1) = 0 \][/tex]

The second statement is True.

3. Reservoir A releases less water than Reservoir B over 1 week.

Check if [tex]\( f(1) < g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not< -1 \][/tex]

This statement is False.

4. Reservoir A releases the same amount of water as Reservoir B over 1 week.

Check if [tex]\( f(1) = g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 = -1 \][/tex]

This statement is True.

5. Reservoir A releases more water than Reservoir B over 1 week.

Check if [tex]\( f(1) > g(1) \)[/tex]:
[tex]\[ f(1) = -1 \][/tex]
[tex]\[ g(1) = -1 \][/tex]
[tex]\[ -1 \not> -1 \][/tex]

This statement is False.

To summarize:
- [tex]\( h(x) = -2x^2 - 13x + 6 \)[/tex] is False.
- [tex]\( h(x) = -2x^2 - x + 3 \)[/tex] is True.
- Reservoir A releases less water than Reservoir B over 1 week is False.
- Reservoir A releases the same amount of water as Reservoir B over 1 week is True.
- Reservoir A releases more water than Reservoir B over 1 week is False.
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