using  a graphing utility to graph the function and finds its domain and range q(1) f(x)=underrootx-1

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.

using a graphing utility to graph the function and find its domain and range q(1) f(x)=underrootx-1

Understand how to graph the square root function f(x) = √(x - 1), determine its domain and range, and evaluate f(1). This detailed explanation covers algebraic concepts, such as the domain of square root functions, suitable for students in Grades 9-11.

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