https://file.aiquickdraw.com/mathbot/file/17265055297126ub19ds2.jpg

The problem in the image is written in Indonesian and asks to determine the domain and range of the function:

\[
f(x) = \sqrt{x^2 - 2x + 5}
\]

### Step 1: Domain of the Function
The domain of a square root function is defined where the expression inside the square root is non-negative. Thus, we need to ensure that:

\[
x^2 - 2x + 5 \geq 0
\]

Now, solve for when the quadratic expression is greater than or equal to zero.

The discriminant of the quadratic expression is:

\[
\Delta = (-2)^2 - 4(1)(5) = 4 - 20 = -16
\]

Since the discriminant is negative, the quadratic expression \( x^2 - 2x + 5 \) has no real roots and is always positive. Therefore, the expression inside the square root is always non-negative for all real \(x\).

Thus, the domain of the function is:

\[
\text{Domain} = (-\infty, \infty)
\]

### Step 2: Range of the Function
Since the expression \( x^2 - 2x + 5 \) is always positive and its minimum value can be found by completing the square:

\[
x^2 - 2x + 5 = (x - 1)^2 + 4
\]

The minimum value of \( (x - 1)^2 \) is 0, which occurs at \(x = 1\). Therefore, the minimum value of \( f(x) \) is:

\[
f(1) = \sqrt{0 + 4} = 2
\]

Thus, the range of the function is:

\[
\text{Range} = [2, \infty)
\]

### Final Answer:
- Domain: \( (-\infty, \infty) \)
- Range: \( [2, \infty) \)

Let me know if you want more details or have any questions! Here are five related questions for practice:

1. How do you find the domain of a function with a square root?
2. What is the role of the discriminant in determining the sign of a quadratic expression?
3. How does completing the square help in finding the minimum value of a quadratic function?
4. What is the general approach to finding the range of a function involving a square root?
5. Can a quadratic expression with a negative discriminant ever be negative?

**Tip:** When solving square root functions, always check the expression inside the root to ensure it is non-negative!

Tentukan domain dan range dari fungsi f(x) = √(x² - 2x + 5).

Learn how to find the domain and range of the function f(x) = √(x² - 2x + 5). This problem involves key mathematical concepts such as functions, quadratic equations, and square roots. Suitable for high school students in Grades 10-12.

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