Question

Write the given function as the composite of two functions, neither of which is the identity function, f(x) = x. f(x) = cubed root of x² + 2

Answer 1

To write the function as the composite of two functions, one function g(x) is defined as the input squared plus two, g(x) = x² + 2, and another function h(x) is the cube root function, h(x) = ∛x. Composing these, h(g(x)), results in the original function f(x) = ∛(x² + 2).

Step-by-step explanation:

To write the function f(x) = ∛x² + 2 as the composite of two functions, neither of which is the identity function, we need to decompose it into two separate functions that when combined, will give us the original function. Let's define the first function, g(x), as g(x) = x² + 2. This function essentially squares the input and then adds 2 to the result. The second function, h(x), could then be the cube root function, defined as h(x) = ∛x. Now, if we compose these two functions where h(x) is the outer function and g(x) is the inner function, the composition h(g(x)), we get:

• g(x) = x² + 2
• h(x) = ∛x
• f(x) = h(g(x)) = ∛(x² + 2)

So, we have successfully expressed f(x) as the composite of two functions g(x) and h(x), which together replicate the original function.