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: f(x) = √(x^(2/3)).

Step-by-step solution

Write down the composition formula.

The composition formula is given by (g ∘ f)(x) = g(f(x)). We are given (g ∘ f)(x) = x^(2/3) + 3.

Substitute g(x) into the composition formula

Replace g(x) with x^2 + 3 in the composition formula: g(f(x)) = (f(x))^2 + 3.

Set the equation equal to the given composition.

We know g(f(x)) should be equal to x^(2/3) + 3, so we set the equation equal to the given composition: (f(x))^2 + 3 = x^(2/3) + 3.

Solve for f(x).

Since both sides of the equation have the same constant term (+3), we can subtract 3 from both sides to isolate f(x): (f(x))^2 = x^(2/3). Now, we need to take the square root of both sides to solve for f(x): f(x) = ±√(x^(2/3)).

Choose the function f(x).

Since we are looking for a single-valued function, we should choose one of the two options. Here, we'll choose the positive square root for f(x): f(x) = √(x^(2/3)). So the function f(x) that produces the given composition is: f(x) = √(x^(2/3)).