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!