Question

Given ( g = (-7, -4), (0, 4), (1, 7), (2, 0), (7, 1) ) and ( h(x) = -3x - 4 ),

What is ( g^-1(1) )?

A. ( g^-1(1) = -7 )

B. ( g^-1(1) = 0 )

C. ( g^-1(1) = 1 )

D. ( g^-1(1) = 2 )

What is ( h^-1(x) )?

A. ( h^-1(x) = -frac1/3(x+4) )

B. ( h^-1(x) = -frac1/3(x-4) )

C. ( h^-1(x) = -frac1/3(x+7) )

D. ( h^-1(x) = -frac1/3(x-7) )

What is ( (h^-1 circ h)(-4) )?

A. ( (h^-1 circ h)(-4) = -4 )

B. ( (h^-1 circ h)(-4) = 0 )

C. ( (h^-1 circ h)(-4) = -7 )

D. ( (h^-1 circ h)(-4) = -1 )

Answer 1

To determine g-1(1), we identify which x-coordinate in g corresponds to the y-coordinate 1, which is 7, making the answer D. (g-1(1) = 2). The inverse function h-1(x) is found by algebraically solving for x, leading to B. h-1(x) = -1/3(x + 4). Lastly, ((h-1 ∘ h)(-4)) results in -4, making the answer A.

Step-by-step explanation:

Let's address each part of the question step by step.

Part 1: To find (g-1(1)), we look for which x-coordinate in function g corresponds to a y-coordinate of 1. Looking at the given pairs in g, we find the pair (7, 1). This means that g-1(1) = 7, so the correct answer is D. (g-1(1) = 2), since that is the x-value that corresponds to the y-value of 1.

Part 2: To find the function h-1(x), we need to solve the equation y = -3x - 4 for x. This gives us x = -(y + 4) / 3. Therefore, the correct inverse function is B. h-1(x) = -1/3(x + 4).

Part 3: The composition ((h-1 ∘ h)(-4)) means we first apply h to -4, and then apply h-1 to the result. We have h(-4) = -3(-4) - 4 = 12 - 4 = 8. Then applying h-1, we get h-1(8) = -1/3(8 + 4) = -1/3(12) = -4. Thus, ((h-1 ∘ h)(-4)) = -4, and the correct answer is A.