whats the inverse of f\left(x\right)=x^{2}+4

To find the inverse of the function \( f(x) = x^2 + 4 \), follow these steps:

1. **Express \( f(x) \) as \( y \):**
   \[ y = x^2 + 4 \]

2. **Swap \( x \) and \( y \) to find the inverse:**
   \[ x = y^2 + 4 \]

3. **Solve for \( y \):**
   \[ x - 4 = y^2 \]
   \[ y = \pm\sqrt{x - 4} \]

Since the original function \( f(x) = x^2 + 4 \) is not one-to-one over the entire set of real numbers (it fails the horizontal line test), its inverse will not be a function unless we restrict the domain. Typically, we restrict the domain to \( x \ge 0 \) or \( x \le 0 \). Let's consider the restriction \( x \ge 0 \):

4. **With the restriction \( x \ge 0 \), the inverse function is:**
   \[ f^{-1}(x) = \sqrt{x - 4} \]

For completeness, if the domain of \( f(x) \) were restricted to \( x \le 0 \), the inverse function would be:
   \[ f^{-1}(x) = -\sqrt{x - 4} \]

Thus, the inverse function of \( f(x) = x^2 + 4 \) is:
\[ f^{-1}(x) = \sqrt{x - 4} \] if \( x \ge 0 \).

If you have any questions or need further clarification, let me know! Here are five more questions you might consider:

1. How do you determine if a function has an inverse?
2. What are the properties of inverse functions?
3. How do you find the inverse of a function involving radicals?
4. How do domain restrictions affect the inverse of a function?
5. What are some common mistakes to avoid when finding the inverse of a function?

**Tip:** Always check if the function is one-to-one (passes the horizontal line test) before finding its inverse. If not, consider restricting the domain.

Discover how to find the inverse of the function f(x) = x^2 + 4 with a detailed step-by-step solution. This problem covers function inverses, quadratic functions, and domain restriction, essential for advanced high school students. Learn how domain restrictions affect the inverse function.

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