f(x)=4x², find the inverse of the quadratic function



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:
\[
f^{-1}(x) = \frac{\sqrt{x}}{2}
\]

Let me know if you want more details or have any questions.

### 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.

f(x)=4x², find the inverse of the quadratic function

Learn how to find the inverse of the quadratic function f(x) = 4x². This detailed guide explains the steps, formulas, and key concepts involved, such as domain restrictions and solving for the inverse. Ideal for students in Grades 10-12.

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