The inverse of this function is:

\large f\left(x\right)=4x^2-8

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**.

The inverse of this function is: f(x) = 4x^2 - 8

Learn how to find the inverse of the function f(x) = 4x^2 - 8. This solution explains the step-by-step process, domain restrictions, and considerations when dealing with quadratic functions. Suitable for students in Grades 10-12.

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