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

Answer

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

### Step 1: Identify the type of function
The given function \( f(x) = x^2 - 2x - 3 \) is a polynomial function of degree 2 (a quadratic function).

### Step 2: Find the domain
The domain of a polynomial function is all real numbers. Therefore, we can state:
\[ \text{Domain} = \mathbb{R} \quad \text{(all real numbers)} \]

### Step 3: Find the range
To find the range of the quadratic function, we can complete the square.

1. Start with the function:
\[ f(x) = x^2 - 2x - 3 \]

2. To complete the square, we can rewrite the quadratic part:
\[ f(x) = (x^2 - 2x) - 3 \]

3. Now, we complete the square for \( x^2 - 2x \):
- Take half of the coefficient of \( x \) (which is -2), square it, and add and subtract it inside the equation:
\[ f(x) = (x^2 - 2x + 1 - 1) - 3 \]
\[ = (x - 1)^2 - 1 - 3 \]
\[ = (x - 1)^2 - 4 \]

4. Now, we have:
\[ f(x) = (x - 1)^2 - 4 \]

5. The expression \( (x - 1)^2 \) is always non-negative (i.e., \( \geq 0 \)). The minimum value occurs when \( x - 1 = 0 \) (or \( x = 1 \)):
\[ \text{Minimum value of } f(x) = 0 - 4 = -4 \]

6. As \( x \) approaches positive or negative infinity, \( f(x) \) will also approach infinity. Thus, the function can take values from its minimum to infinity:
\[ \text{Range} = [-4, \infty) \]

### Final Answer:
- **Domain**: \( \mathbb{R} \) (all real numbers)
- **Range**: \( [-4, \infty) \)