Given the function:

$f(x) = - (x - 1)^2 + 1$

we are tasked with determining the domain and range of the function $y = \sqrt{f(x)}$.

Step 1: Identify the domain of $f(x)$

The function $f(x) = - (x - 1)^2 + 1$ is a quadratic function. The domain of a quadratic function is all real numbers, so initially, we have:

$\text{Domain of } f(x) = (-\infty, \infty)$

Step 2: Identify the condition for the square root

Since the expression involves a square root, for $y = \sqrt{f(x)}$ to be valid, we need the value inside the square root, $f(x)$, to be non-negative (i.e., $f(x) \geq 0$).

Thus, we need to find where $f(x) \geq 0$. Let's solve this inequality:

$- (x - 1)^2 + 1 \geq 0$

Rearranging terms:

$- (x - 1)^2 \geq -1$

Multiply both sides by -1 (which reverses the inequality):

$(x - 1)^2 \leq 1$

Step 3: Solve the inequality

Now, we solve $(x - 1)^2 \leq 1$:

$-1 \leq x - 1 \leq 1$

Adding 1 to all parts of the inequality:

$0 \leq x \leq 2$

Thus, the domain of $y = \sqrt{f(x)}$ is $[0, 2]$, as $f(x)$ is non-negative only within this interval.

Step 4: Find the range of $y = \sqrt{f(x)}$

To find the range, we first analyze the possible values of $f(x)$ on the domain $[0, 2]$.

At the endpoints of the domain:

• When $x = 0$, $f(0) = - (0 - 1)^2 + 1 = -1 + 1 = 0$
• When $x = 2$, $f(2) = - (2 - 1)^2 + 1 = -1 + 1 = 0$

The maximum value of $f(x)$ occurs at $x = 1$, where $f(1) = - (1 - 1)^2 + 1 = 1$.

Since $f(x)$ decreases as $x$ moves away from 1 (in both directions), $f(x)$ reaches values between 0 and 1.

Thus, $y = \sqrt{f(x)}$ will take values from 0 to 1. The range of $y$ is:

$\text{Range of } y = [0, 1]$

Final Answer:

• Domain of $y = \sqrt{f(x)}$: $[0, 2]$
• Range of $y = \sqrt{f(x)}$: $[0, 1]$

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Would you like further details or clarification on any step?

Here are some related questions you might find interesting:

1. How do you find the vertex of a quadratic function?
2. What does the transformation $f(x) = -(x - 1)^2 + 1$ represent graphically?
3. How can you graph the function $y = \sqrt{f(x)}$?
4. What happens to the range of $y = \sqrt{f(x)}$ if the expression inside the square root is modified?
5. How does the square root affect the shape of the graph of a function?

Tip: When working with square roots of functions, always ensure the expression inside the root is non-negative to keep the function real-valued.