Question

Find two nontrivial functions f(x) and g(x) so that f(g(x)) = (-6-3x)⁶

Answer 1

Let
`\( f(x) = (-6-x)^2 \) and \( g(x) = x^3 \)`, then
`\( f(g(x)) = (-6-3x)^6 \).`

Step-by-step explanation:

To find nontrivial functions f(x) and g(x) such that
`\( f(g(x)) = (-6-3x)^6 \),`we need to carefully construct the compositions. Let
`\( g(x) = x^3 \)` as a cube function, and
`\ f(x) = (-6-x)^2 \)`as a quadratic function. When we substitute g(x) into f(x) , we get
`\( f(g(x)) = (-6-3x)^6 \)`. The choice of powers in g(x) and f(x) is essential for achieving the desired result.

The composition f(g(x)) implies that we first apply g(x) and then f(x) to the result. In this case, cubing x with g(x) is followed by squaring the quantity -6-x with f(x) , resulting in the given expression
`\( (-6-3x)^6 \).`

It's crucial to select functions that allow for a meaningful composition, and in this example, the cube and square functions are chosen strategically to achieve the specified result. This demonstrates the importance of understanding function composition and carefully selecting functions to meet given criteria.