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 is \(f(x) = \pm x^2\).

Step-by-step solution

Write the composition for \((g \circ f)(x)\)

We are given that \((g \circ f)(x) = x^4 + 3\) and \(g(x)=x^2+3\). We can represent the composition as: $$g(f(x))=x^4+3$$

Solve for the function \(f(x)\)

Now we will replace \(g(x)\) with its definition in terms of \(x\): $$g(f(x))=x^4+3 \Rightarrow (f(x))^2 +3 = x^4 + 3$$ Now, we need to solve for the function \(f(x)\). Subtracting \(3\) from both sides of the equation: $$(f(x))^2 = x^4$$ Taking the square root of both sides: $$f(x) = \pm\sqrt{x^4}$$ This is our function \(f(x)\), and we can represent it as: $$f(x) = \pm x^2$$