Question

Given f(x) = √x² +3, determine p(x), q(x) and r(x) such that f(x) =p (q(r(x))). You can proceed to determine the functions p(x), q(x), and r(x) to express f(x) as a composition of these functions.

Answer 1

The function f(x) is broken down into p(x) = √x, q(x) = x + 3, and r(x) = x², which are composed to form the original function as f(x) = p(q(r(x))).

Step-by-step explanation:

To express the function f(x) = √x² +3 as a composition of three functions such that f(x) = p(q(r(x))), we can break down the computation into simpler steps. As an example:

• Let r(x) = x² to represent the squaring part of our innermost function.
• Next, let q(x) = x + 3 to account for the addition of 3 that follows the squaring in our original function.
• Finally, let p(x) = √x where we take the square root, completing the overall composition to achieve the original function.

This sequence of operations will yield the original function: f(x) = p(q(r(x))) = p(q(x²)) = p(x² + 3) = √(x² + 3).

In the context of quadratic functions and their solutions, understanding compositions of functions is essential. Given a certain operation like squaring or addition within a function, you can decompose it into constituent functions that are easier to manage, which is helpful for various mathematical analyses, including equilibrium problems.