To find the inverse of the function $f(x) = 4x^2 - 8$, follow these steps:

Step 1: Replace $f(x)$ with $y$

$y = 4x^2 - 8$

Step 2: Swap $x$ and $y$

$x = 4y^2 - 8$

Step 3: Solve for $y$

1. Add 8 to both sides: $x + 8 = 4y^2$
2. Divide by 4: $\frac{x + 8}{4} = y^2$
3. Take the square root of both sides: $y = \pm \sqrt{\frac{x + 8}{4}}$

Step 4: Consider Domain Restrictions

Since the original function $f(x) = 4x^2 - 8$ is a quadratic function, it is not one-to-one unless we restrict its domain.

• If $f(x)$ is given with $x \geq 0$, then we take the positive root: $f^{-1}(x) = \sqrt{\frac{x + 8}{4}}$
• If $x \leq 0$, we take the negative root: $f^{-1}(x) = -\sqrt{\frac{x + 8}{4}}$

Let me know if you have any questions or want further clarification!

Related Questions:

1. How do you determine if a function has an inverse?
2. What happens if a function is not one-to-one?
3. How do you graph a function and its inverse?
4. Can you restrict the domain of any function to make it invertible?
5. What is the inverse of $g(x) = ax^2 + b$ in general?

Tip:

A function has an inverse only if it is one-to-one, meaning it passes the horizontal line test (each $y$-value is produced by only one $x$-value). If not, we must restrict its domain.