https://file.aiquickdraw.com/mathbot/file/1727354235494v07ozf4c.jpg

The function in the image is \( f(x) = x^2 + 3 \). Let's now find its inverse.

### Steps to find the inverse of a function:
1. Start with \( f(x) = y = x^2 + 3 \).
2. Swap \( x \) and \( y \): 
   \[ x = y^2 + 3 \]
3. Solve for \( y \):
   \[ x - 3 = y^2 \]
   \[ y = \pm \sqrt{x - 3} \]

So, the inverse function is:

\[
f^{-1}(x) = \pm \sqrt{x - 3}
\]

Since a quadratic function \( f(x) = x^2 + 3 \) is not one-to-one, we have two possible branches of the inverse: 
- For the positive branch: \( f^{-1}(x) = \sqrt{x - 3} \)
- For the negative branch: \( f^{-1}(x) = -\sqrt{x - 3} \)

Would you like to discuss any specific branch or further details on inverse functions?

Here are 5 follow-up questions:
1. How can we determine the domain of the inverse function?
2. What is the significance of choosing one branch (positive or negative)?
3. Can the original function be restricted to make the inverse a function without the \( \pm \)?
4. How do we verify if a function is invertible?
5. What would happen if the original function had a linear term added?

**Tip:** Inverses of functions are symmetric along the line \( y = x \), so graphing both can help visualize the relationship.

Find the inverse function of f(x) = x^2 + 3

Learn how to find the inverse of the quadratic function f(x) = x^2 + 3. Follow the step-by-step solution involving algebraic manipulation, including dealing with the positive and negative branches of the square root. Suitable for Grades 10-12.

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