Question

Determine wether f(x) = ∛(2x - 3) and g(x) = 1/(2x^4 + 3x), x ≥ 0, are inverse functions?

a) The functions are inverse.
b) The functions are not inverse.

Answer 1

The functions f(x) = ∛(2x - 3) and g(x) = 1/(2x^4 + 3x) are not inverse functions.

Step-by-step explanation:

To determine whether the functions f(x) = ∛(2x - 3) and g(x) = 1/(2x^4 + 3x) are inverse functions, we need to check if their compositions result in the identity function. Let's compose the functions f(g(x)) and g(f(x)).

For f(g(x)), substituting g(x) into f(x), we get ∛(2(1/(2x^4 + 3x)) - 3) = ∛((1/x^4) + (3/x) - 3). This is not equal to x because the expression inside the cube root contains negative exponents. Therefore, f(g(x)) is not equal to x.

For g(f(x)), substituting f(x) into g(x), we get 1/(2(∛(2x - 3))^4 + 3(∛(2x - 3))) = 1/(2∛(2x - 3)^4 + 3∛(2x - 3)). This is also not equal to x. Hence, g(f(x)) is not equal to x, and therefore, the functions f(x) and g(x) are not inverse functions.