Find the inverse of the function.
​f(x)equalsminusxsquaredplus 19​, xgreater than or equals0

To find the inverse of the function \( f(x) = -x^2 + 19 \) where \( x \geq 0 \), we will follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
So, the equation becomes:
\[
y = -x^2 + 19
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we swap \( x \) and \( y \):
\[
x = -y^2 + 19
\]

### Step 3: Solve for \( y \)
Now, we solve for \( y \):

\[
x - 19 = -y^2
\]
\[
y^2 = 19 - x
\]
\[
y = \sqrt{19 - x}
\]

(Note: Since \( x \geq 0 \), we take the positive square root because the original function specifies that \( x \geq 0 \).)

### Step 4: Write the inverse function
Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt{19 - x}
\]

### Final Answer:
The inverse of \( f(x) = -x^2 + 19 \) for \( x \geq 0 \) is:
\[
f^{-1}(x) = \sqrt{19 - x}
\]

Let me know if you'd like any further clarification or if you have more questions!

### Related Questions:
1. How would the inverse change if \( x \) were less than 0?
2. How can you graph both the function and its inverse?
3. What is the domain and range of the inverse function?
4. What is the significance of the restriction \( x \geq 0 \) in the context of finding the inverse?
5. How would you verify if two functions are true inverses?

### Tip:
When finding the inverse of a quadratic function, always remember to consider the domain and range restrictions to ensure that the inverse is a function.

Find the inverse of the function f(x) = -x^2 + 19, x ≥ 0

Learn how to find the inverse of the quadratic function f(x) = -x^2 + 19 for x ≥ 0 with a step-by-step solution. Understand the domain and range restrictions, and explore the key concepts of inverse functions and square root transformations. Suitable for students in Grades 10-12.

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