Question

Given f(x), find g(x) and h(x) such that f(x)=g(h(x)) and neither g(x) nor h(x) is solely x.

f(x)= ∛-3x+4-5

Answer 1

Decompose the given function into two functions g(x) and h(x) by identifying h(x) as the operation inside the cube root, which is h(x)=-3x+4, and then g(x) as the outer operation, which is g(x)=∛x-5.

Step-by-step explanation:

The problem requires us to decompose the function f(x) = ∛(-3x + 4) - 5 into two functions g(x) and h(x) such that f(x) = g(h(x)), and neither g(x) nor h(x) are just x. To do this, we need to identify an inner function h(x) and an outer function g(x). A natural way to do this is to let the inside of the cube root and subsequent subtraction be our h(x).

Let us define h(x) = -3x + 4. Here, h(x) represents the operation inside the cube root. Next, we need to apply the cube root and then subtract 5, which will serve as our g(x).

Therefore, we define g(x) = ∛x - 5. By using h(x), we address the operation inside the cube root and with g(x), we perform the cube root and subtract 5.

So, f(x) = g(h(x)) becomes f(x) = g(-3x + 4) = ∛(-3x + 4) - 5, which matches the original function, therefore making our choice for g(x) and h(x) valid.