Question

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(f \circ g)(x)=x^{4}+6 x^{2}+9$$

Short answer

Answer: The function \(f(x)\) is \(f(x)=x^{4}+6x^{2}+9\).

Step-by-step solution

Write the given information

We have the following given information: - \(g(x)=x^{2}+3\) - \((f\circ g)(x)=x^{4}+6x^{2}+9\)

Replace \((f\circ g)(x)\) with \(f(g(x))\)

To better understand the relationship between \(f(x)\) and \(g(x)\), let's rewrite \((f\circ g)(x)\) as \(f(g(x))\). This means: $$f(g(x))=x^{4}+6x^{2}+9$$

Substitute \(g(x)\) into the equation

Since we know that \(g(x)=x^{2}+3\), we can substitute this expression into the equation and solve for \(f\): $$f(x^{2}+3)=x^{4}+6x^{2}+9$$

Find the inverse function of \(g(x)\)

Since we want to find \(f(x)\), we need to get rid of the \(x^{2}+3\) in the argument of the function \(f\). We can do this by finding the inverse function of \(g(x)\), let's call it \(g^{-1}(x)\). We know that: $$g(x)=x^{2}+3$$ Now, let's find the inverse. Swap \(x\) and \(y\) in the equation: $$x=y^{2}+3$$ Now, solve for \(y\): $$y^{2}=x-3$$ $$y=\sqrt{x-3}$$ So the inverse function \(g^{-1}(x)\) is: $$g^{-1}(x)=\sqrt{x-3}$$

Use the inverse function to find \(f(x)\)

We can now use the inverse function to find \(f(x)\). Replace the argument \(x^{2}+3\) in the equation \(f(x^{2}+3)=x^{4}+6x^{2}+9\) with \(g^{-1}(x)\): $$f(\sqrt{x-3})=x^{4}+6x^{2}+9$$ Now that we have the equation in terms of \(f(x)\), the function \(f(x)\) that we were looking for is: $$f(x)=x^{4}+6x^{2}+9$$