The given function is $f(x) = 4x^2$. To find the inverse of a quadratic function, we need to follow these steps:

Steps:

1. Replace $f(x)$ with $y$: $y = 4x^2$
2. Switch $x$ and $y$ to solve for $y$ (the inverse function): $x = 4y^2$
3. Solve for $y$: Divide both sides by 4: $\frac{x}{4} = y^2$ Now, take the square root of both sides: $y = \pm \sqrt{\frac{x}{4}} = \pm \frac{\sqrt{x}}{2}$ So, the inverse function is: $f^{-1}(x) = \pm \frac{\sqrt{x}}{2}$

Important Notes:

Since quadratic functions are not one-to-one (they fail the horizontal line test), the inverse function is not a true function unless we restrict the domain of the original function. If the domain is restricted, for example, to $x \geq 0$, then the inverse becomes: $f^{-1}(x) = \frac{\sqrt{x}}{2}$

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5 Related Questions:

1. How do you find the inverse of a cubic function?
2. What happens to the graph of the function when finding its inverse?
3. How do domain restrictions affect the inverse of quadratic functions?
4. Can all quadratic functions have an inverse? Why or why not?
5. How can you verify that two functions are inverses of each other?

Tip: When solving for the inverse of a quadratic, always check if the domain needs restriction for the inverse to be a valid function.