To graph the function $f(x) = \sqrt{x - 1}$, and find its domain and range, we can analyze the function step by step.

1. Domain of the Function:

The domain of a function consists of all the possible input values (x-values) for which the function is defined. Since the square root function $\sqrt{x - 1}$ is only defined when the value inside the square root is non-negative, we have the following condition: $x - 1 \geq 0$ Solving for $x$: $x \geq 1$ Thus, the domain of the function is: $\text{Domain: } [1, \infty)$ This means that $f(x)$ is defined for all $x \geq 1$.

2. Range of the Function:

The range of the function consists of all the possible output values (y-values) of the function. Since the square root function only gives non-negative results, and $f(x) = \sqrt{x - 1}$ outputs 0 when $x = 1$ and increases as $x$ increases, the range is: $\text{Range: } [0, \infty)$

3. Graphing the Function:

The graph of $f(x) = \sqrt{x - 1}$ starts at the point $(1, 0)$ and increases slowly as $x$ increases, since the square root function grows slower for larger values of $x$.

4. Evaluating $q(1)$:

Given $f(x) = \sqrt{x - 1}$, we are asked to find $f(1)$: $f(1) = \sqrt{1 - 1} = \sqrt{0} = 0$ So, $f(1) = 0$.

Summary:

• Domain: $[1, \infty)$
• Range: $[0, \infty)$
• $f(1) = 0$

Would you like me to graph this function or provide more details?

Related Questions:

1. How do we determine the domain of more complex square root functions?
2. What happens if the function includes a vertical shift or a horizontal shift?
3. How do you find the domain of functions involving square roots and fractions?
4. How does a graphing utility help visualize the domain and range?
5. What are the key differences between square root functions and polynomial functions?

Tip:

Always check the inside of the square root for non-negative values to find the domain of a square root function.