Let's break down the function you provided:

The function is given as:

$y = \sqrt{x - 1} + 3$

Domain:

The domain is the set of all possible $x$-values for which the function is defined. Since you have a square root function, the expression inside the square root must be non-negative (greater than or equal to zero) for the function to be real and valid.

For $\sqrt{x - 1}$, the expression inside the square root must satisfy:

$x - 1 \geq 0$

Solving this inequality:

$x \geq 1$

So, the domain of the function is:

$x \geq 1 \quad \text{or} \quad [1, \infty)$

Range:

To find the range, we need to determine the possible values of $y$ (the output). The square root function $\sqrt{x - 1}$ is always non-negative (i.e., $\sqrt{x - 1} \geq 0$) and it increases as $x$ increases. The smallest value of $\sqrt{x - 1}$ occurs when $x = 1$, which gives:

$\sqrt{1 - 1} = 0$

Therefore, when $x = 1$, $y = 0 + 3 = 3$. As $x$ increases, $\sqrt{x - 1}$ increases, which means $y$ increases. So, the range starts from 3 and goes to infinity.

Thus, the range of the function is:

$y \geq 3 \quad \text{or} \quad [3, \infty)$

Final Answer:

• Domain: $[1, \infty)$
• Range: $[3, \infty)$

Would you like more details or have any questions about this?

Here are some related questions:

1. How would you find the domain and range of a function involving a rational expression?
2. Can the domain and range be determined graphically? How?
3. What happens if we add a negative sign inside the square root function?
4. How would transformations, like shifting or scaling, affect the domain and range?
5. How do you handle composite functions for domain and range?

Tip: Always check the conditions (like non-negative values for square roots) when determining the domain and range of functions involving radicals.