Question

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(g \circ f)(x)=x^{4}+3$$

Short answer

Answer: The function f(x) that satisfies the given composition is f(x) = x^2.

Step-by-step solution

Analyze the given function g(x) and composition

Given function: \(g(x) = x^2 + 3\) Composition: \((g\circ f)(x) = x^4 + 3\) Recall that the composition \((g\circ f)(x)=g(f(x))\). Our task is to find \(f(x)\) such that when it is substituted into \(g(x)\), we obtain the composition \((g\circ f)(x)\).

Substitute f(x) into g(x)

Let's substitute \(f(x)\) into \(g(x)\). We have: \(g(f(x)) = (f(x))^2 + 3\) Now, we want to find \(f(x)\) such that: \((f(x))^2 + 3 = x^4 + 3\)

Analyze the equation

We can see that if the two expressions are equal, then the following must be true: \((f(x))^2 = x^4\)

Solve for f(x)

To find \(f(x)\), we need to take the square root of both sides of the equation: \(\sqrt{(f(x))^2} = \sqrt{x^4}\) This gives us: \(f(x) = x^2\) (Note that we only take the positive square root, as \(f(x)\) is a function and thus must have a single output for each input)

Check the solution

Finally, let's check that our proposed solution for \(f(x) = x^2\) actually works: \(g(f(x)) = g(x^2) = (x^2)^2 + 3 = x^4 + 3\) So, the function \(f(x)\) that produces the given composition is indeed: \(f(x) = x^2\)