Given the function defined by f (x) = x² - 4. Write the domain and range of f. Graph f by making a table of ordered pairs.

### Domain and Range of the Function

The function given is \( f(x) = x^2 - 4 \).

1. **Domain**: The domain of a function includes all possible values of \( x \) for which the function is defined. Since \( f(x) = x^2 - 4 \) is a quadratic function (a polynomial of degree 2), it is defined for all real numbers. 
   
   Therefore, the domain of \( f(x) \) is:
   \[
   \text{Domain} = (-\infty, \infty)
   \]

2. **Range**: The range of a function is the set of all possible values of \( f(x) \) as \( x \) varies over the domain. For a quadratic function of the form \( f(x) = ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards, and the minimum value occurs at the vertex. In this case, \( a = 1 \), so the parabola opens upwards.

   The vertex of the parabola can be found using the formula \( x = -\frac{b}{2a} \). Here, \( b = 0 \), so:
   \[
   x = 0
   \]
   Substituting \( x = 0 \) into the function:
   \[
   f(0) = 0^2 - 4 = -4
   \]
   Thus, the minimum value of \( f(x) \) is \(-4\), and since the parabola opens upwards, the range is:
   \[
   \text{Range} = [-4, \infty)
   \]

### Graphing the Function Using a Table of Ordered Pairs

To graph the function \( f(x) = x^2 - 4 \), we can create a table of values by choosing different values of \( x \) and calculating the corresponding values of \( f(x) \).

| \( x \) | \( f(x) = x^2 - 4 \) | Ordered Pair (x, f(x)) |
|:------:|:--------------------:|:----------------------:|
|  -3    |      \( 9 - 4 = 5 \)   |       (-3, 5)          |
|  -2    |      \( 4 - 4 = 0 \)   |       (-2, 0)          |
|  -1    |      \( 1 - 4 = -3 \)  |       (-1, -3)         |
|   0    |      \( 0 - 4 = -4 \)  |       (0, -4)          |
|   1    |      \( 1 - 4 = -3 \)  |       (1, -3)          |
|   2    |      \( 4 - 4 = 0 \)   |       (2, 0)           |
|   3    |      \( 9 - 4 = 5 \)   |       (3, 5)           |

### Graph Interpretation

- The vertex of the parabola is at the point (0, -4).
- The parabola opens upwards.
- The points (-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), and (3, 5) are symmetric about the y-axis.

You can plot these points on a coordinate plane and connect them with a smooth curve to visualize the graph of the function.

Would you like more details or have any questions?

### Related Questions:
1. What are the key features of the graph of a quadratic function?
2. How do you find the vertex of a parabola given its equation?
3. How does the value of "a" in a quadratic function \( f(x) = ax^2 + bx + c \) affect the graph?
4. How can the graph of \( f(x) = x^2 - 4 \) be transformed into the graph of \( g(x) = x^2 + 4 \)?
5. What are the roots of the function \( f(x) = x^2 - 4 \) and how do they relate to the graph?

**Tip**: When graphing a quadratic function, always find the vertex first, as it provides a useful reference point for plotting the rest of the graph.

Explore the domain and range of the quadratic function f(x) = x² - 4. Learn how to graph the function by creating a table of ordered pairs. Understand the properties of quadratic functions and their graphs. Suitable for high school students.

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