Question

Use composition to determine if G(x) or H(x) is the inverse of F(x) across the domain x≥ -7.

F(x)=√(x+7)
G(x)=(x-7)²
H(x)=x²+7

A. Both functions are inverses of F(x).
B. G(x) is the inverse of F(x).
C. Neither function is the inverse of F(x).
D. H(x) is the inverse of F(x).

Answer 1

Upon testing both G(x) and H(x) through composition with F(x), neither composition results in the identity function, which implies that neither G(x) nor H(x) is the inverse of F(x). Therefore, the correct option is C: Neither function is the inverse of F(x).

Step-by-step explanation:

To determine if G(x) or H(x) is the inverse of F(x) across the domain x≥ -7, we need to check whether composing each function with F(x) results in the identity function over the specified domain. The function F(x) = √(x+7) has two potential inverses to check: G(x) = (x-7)^2 and H(x) = x^2 + 7.

For G(x), let's find F(G(x)):

F(G(x)) = F((x-7)^2) = √((x-7)^2 + 7) = √(x^2 - 14x + 49 + 7) = √(x^2 - 14x + 56).

This function is not the identity function, since we do not simplify to x; thus, G(x) is not the inverse of F(x).

Now for H(x), let's find F(H(x)):

F(H(x)) = F(x^2 + 7) = √((x^2 + 7) + 7) = √(x^2 + 14) which is also not equal to x; therefore, H(x) is also not the inverse of F(x).

Given these compositions, we can conclude that option C: Neither function is the inverse of F(x) is correct.