The given function is:

$f(x) = \sqrt{x^2 - 4x + 5}$

To understand the behavior and analyze this function, it is helpful to simplify the expression under the square root. Let's rewrite the quadratic expression inside the square root in a more convenient form by completing the square.

Step 1: Completing the square

The quadratic expression is $x^2 - 4x + 5$. To complete the square, we focus on the first two terms $x^2 - 4x$.

1. Take half of the coefficient of $x$, which is $-4$, and divide it by 2 to get $-2$.
2. Square $-2$ to get $4$.

Now, rewrite $x^2 - 4x$ as:

$x^2 - 4x = (x - 2)^2 - 4$

Thus, the original quadratic expression becomes:

$x^2 - 4x + 5 = (x - 2)^2 - 4 + 5 = (x - 2)^2 + 1$

Step 2: Substituting back into the original function

Now, we substitute this back into the original function:

$f(x) = \sqrt{(x - 2)^2 + 1}$

Step 3: Interpretation

The expression $f(x) = \sqrt{(x - 2)^2 + 1}$ shows that the function represents the distance from any point $x$ on the real line to the point $x = 2$ along the x-axis, but shifted vertically by 1. The minimum value of the expression inside the square root is $1$, which occurs when $x = 2$. Therefore, the minimum value of $f(x)$ is:

$f(2) = \sqrt{(2 - 2)^2 + 1} = \sqrt{1} = 1$

Step 4: Domain of the function

Since the expression inside the square root, $(x - 2)^2 + 1$, is always positive for all real $x$, the function is defined for all real numbers. Thus, the domain of $f(x)$ is:

$\text{Domain of } f(x): \mathbb{R}$

Step 5: Range of the function

Since the minimum value of $f(x)$ is $1$ and there is no upper bound (because $(x - 2)^2$ can grow infinitely large), the range of $f(x)$ is:

$\text{Range of } f(x): [1, \infty)$

Final Form of the Function:

The function can be written as:

$f(x) = \sqrt{(x - 2)^2 + 1}$

Questions for Further Exploration:

1. How does the graph of $f(x)$ look like?
2. What are the asymptotes or key features of the graph of $f(x)$?
3. How does the function behave as $x$ approaches infinity or negative infinity?
4. How would the graph of $f(x)$ change if the constant $+1$ inside the square root were replaced with another value?
5. How does completing the square help in analyzing functions involving square roots?

Tip:

When analyzing functions involving square roots, always check for opportunities to complete the square, as it can simplify the expression and make it easier to identify the key properties of the function.