Find the domain and the range of the function `y=f(x)`, where `f(x)` is given by
`sqrt(x^(2)-2x-3)`

Answer

To find the domain and range of the function \( y = f(x) = \sqrt{x^2 - 2x - 3} \), we will follow these steps:

### Step 1: Identify the expression under the square root
The function is defined as:
\[ f(x) = \sqrt{x^2 - 2x - 3} \]
To ensure that \( f(x) \) is a real number, the expression inside the square root must be non-negative:
\[ x^2 - 2x - 3 \geq 0 \]

### Step 2: Factor the quadratic expression
We will factor the quadratic expression \( x^2 - 2x - 3 \):
\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
Thus, we need to solve the inequality:
\[ (x - 3)(x + 1) \geq 0 \]

### Step 3: Determine the critical points
The critical points occur where the expression equals zero:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
These points divide the number line into intervals.

### Step 4: Test the intervals
We will test the sign of \( (x - 3)(x + 1) \) in the intervals determined by the critical points:
1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \):
\[ (-2 - 3)(-2 + 1) = (-5)(-1) = 5 \quad \text{(positive)} \]
2. **Interval \( (-1, 3) \)**: Choose \( x = 0 \):
\[ (0 - 3)(0 + 1) = (-3)(1) = -3 \quad \text{(negative)} \]
3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \):
\[ (4 - 3)(4 + 1) = (1)(5) = 5 \quad \text{(positive)} \]

### Step 5: Write the solution for the inequality
The expression \( (x - 3)(x + 1) \geq 0 \) is satisfied in the intervals:
\[ (-\infty, -1] \cup [3, \infty) \]
Thus, the **domain** of \( f(x) \) is:
\[ \text{Domain: } (-\infty, -1] \cup [3, \infty) \]

### Step 6: Determine the range of the function
Since \( f(x) = \sqrt{x^2 - 2x - 3} \) is a square root function, it will yield non-negative values. The minimum value occurs at the endpoints of the domain:
- At \( x = -1 \):
\[ f(-1) = \sqrt{(-1)^2 - 2(-1) - 3} = \sqrt{1 + 2 - 3} = \sqrt{0} = 0 \]
- At \( x = 3 \):
\[ f(3) = \sqrt{3^2 - 2(3) - 3} = \sqrt{9 - 6 - 3} = \sqrt{0} = 0 \]
As \( x \) approaches \( -\infty \) or \( +\infty \), \( f(x) \) approaches \( +\infty \).

Thus, the **range** of \( f(x) \) is:
\[ \text{Range: } [0, \infty) \]

### Final Answer
- **Domain**: \( (-\infty, -1] \cup [3, \infty) \)
- **Range**: \( [0, \infty) \)