Question

Consider these functions:

f(x) = 16x^2
g(x) = 1/4√x

For x ≥ 0, the value of f(g(x)) is ____(x, 4x, 8x).
For x ≥ 0, the value of g(f(x)) is ____(x, 4x, 8x).
For x ≥ 0, functions f and g ____ inverse functions (are, are not).

Answer 1

The value of f(g(x)) is 4x and the value of g(f(x)) is 8x. Functions f and g are not inverse functions.

Step-by-step explanation:

For x ≥ 0, the value of f(g(x)) is 4x. To find this, substitute g(x) into f(x), so f(g(x)) = 16(g(x))^2. Since g(x) = 1/4√x, substitute this into f(g(x)) to get 16((1/4√x))^2 = 16(1/16x) = 1/x. Simplifying further, we get f(g(x)) = 1/x = 4x.

For x ≥ 0, the value of g(f(x)) is 8x. To find this, substitute f(x) into g(x), so g(f(x)) = 1/4√(f(x)). Since f(x) = 16x^2, substitute this into g(f(x)) to get 1/4√(16x^2) = 1/4(4x) = x. Therefore, g(f(x)) = x = 8x.

For x ≥ 0, functions f and g are not inverse functions. Inverse functions undo each other's calculations, but in this case, the output of f(g(x)) is different from the input x.