Questions: Which functions have a maximum value greater than the maximum of the function g(x)=-(x+3)^2-4 ? Check all that apply. f(x)=-(x+1)^2-2 f(x)=-x+4-5 f(x)=-√(x+2) f(x)=-2x+3 f(x)=-√(x-2)-4

Solution Steps

To determine which functions have a maximum value greater than the maximum of the function \( g(x)=-(x+3)^{2}-4 \), we need to find the maximum value of each function and compare it to the maximum value of \( g(x) \).

1. Find the maximum value of \( g(x) \).
2. Find the maximum value of each given function.
3. Compare these maximum values to the maximum value of \( g(x) \).

The function given is \( g(x) = -(x + 3)^2 - 4 \). This is a downward-opening parabola, and its maximum value occurs at the vertex. The vertex of \( g(x) \) is at \( x = -3 \).

Substituting \( x = -3 \) into \( g(x) \): \[ g(-3) = -(-3 + 3)^2 - 4 = -0 - 4 = -4 \] Thus, the maximum value of \( g(x) \) is \(-4\).

This is also a downward-opening parabola. The vertex is at \( x = -1 \).

Substituting \( x = -1 \) into \( f_1(x) \): \[ f_1(-1) = -(-1 + 1)^2 - 2 = -0 - 2 = -2 \] Thus, the maximum value of \( f_1(x) \) is \(-2\).

This function reaches its maximum value when \( |x + 4| \) is minimized, which occurs at \( x = -4 \).

Substituting \( x = -4 \) into \( f_2(x) \): \[ f_2(-4) = -|-4 + 4| - 5 = -0 - 5 = -5 \] Thus, the maximum value of \( f_2(x) \) is \(-5\).

This function reaches its maximum value when \( \sqrt{x + 2} \) is minimized, which occurs at \( x = -2 \).

Substituting \( x = -2 \) into \( f_3(x) \): \[ f_3(-2) = -\sqrt{-2 + 2} = -\sqrt{0} = 0 \] Thus, the maximum value of \( f_3(x) \) is \( 0 \).

This function reaches its maximum value when \( |2x| \) is minimized, which occurs at \( x = 0 \).

Substituting \( x = 0 \) into \( f_4(x) \): \[ f_4(0) = -|2 \cdot 0| + 3 = -0 + 3 = 3 \] Thus, the maximum value of \( f_4(x) \) is \( 3 \).

This function reaches its maximum value when \( \sqrt{x - 2} \) is minimized, which occurs at \( x = 2 \).

Substituting \( x = 2 \) into \( f_5(x) \): \[ f_5(2) = -\sqrt{2 - 2} - 4 = -\sqrt{0} - 4 = -4 \] Thus, the maximum value of \( f_5(x) \) is \(-4\).

We compare the maximum values of each function to the maximum value of \( g(x) \):

• \( f_1(x) = -2 \) (greater than \(-4\))
• \( f_2(x) = -5 \) (not greater than \(-4\))
• \( f_3(x) = 0 \) (greater than \(-4\))
• \( f_4(x) = 3 \) (greater than \(-4\))
• \( f_5(x) = -4 \) (not greater than \(-4\))

Final Answer