Question

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

Short answer

Answer: The function \(f(x) = x^{\frac{1}{3}}\) produces the given composition \((g \circ f)(x) = x^{\frac{2}{3}} + 3\) with \(g(x) = x^2 + 3\).

Step-by-step solution

Express the composition as \(g(f(x))\)

Given: \((g \circ f)(x)=x^{\frac{2}{3}}+3\). This means \(g(f(x)) = x^{\frac{2}{3}} + 3\).

Solve for \(f(x)\) in terms of g

We know that \(g(x) = x^2 + 3\). Let's say \(f(x)=a\). So we have \(g(a) = a^2 + 3 = x^{\frac{2}{3}} + 3\).

Isolate \(a^2\)

We want to extract \(a\) from this equation to find \(f(x)\). So, we rewrite the equation and subtract \(3\) from both sides: $$a^2 = x^{\frac{2}{3}}$$

Solve for \(a\)

Since we have \(a^2\), to find \(a\) (which is our \(f(x)\)) we just need to take the square root of both sides of the equation: $$\sqrt{a^2} = \sqrt{x^{\frac{2}{3}}}$$ Now, we have: $$a = x^{\frac{1}{3}}$$

Write the final \(f(x)\)

The solution is the function \(f(x) = x^{\frac{1}{3}}\). This function will produce the given composition \((g \circ f)(x)\) when plugged in place of \(x\) in function \(g(x)\): $$(g \circ f)(x) = g\left(x^{\frac{1}{3}}\right) = \left(x^{\frac{1}{3}}\right)^2 + 3 = x^{\frac{2}{3}} + 3$$