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The domain and range of a function are vital to understand its limitations and outputs. The **domain** comprises all possible input values (or x-values) for a function. There are no restrictions on the variable \( x \) for the function \( h(x) = |x| - 3 \). This is typical for absolute value functions unless there are additional terms that limit \( x \), such as fractions or square roots. Therefore, the domain for \( h(x) \) is all real numbers, expressed as

• Domain: \((-\infty, \infty)\)

The **range**, however, is what the function can output (or y-values). The lowest point on the graph of \( h(x) \) is at \( y = -3 \). This value occurs when the absolute value component is zero, which is at the vertex. The V-shape indicates that as \( x \) moves away from 0, \( y \) only gets larger. Hence, the range is from \( -3 \) including all values above that, represented as:

• Range: \([-3, \infty)\)

To graph the absolute value function \( h(x) = |x| - 3 \), start by recognizing its basic shape. Absolute value functions have a distinct V shape. We begin sketching this function from its vertex. Since \( h(x) \) is derived from \( |x| \), the primary modification is a downward shift by 3 units.
Start at the vertex located at \( (0, -3) \). The graph then extends upward to the left and right. For better accuracy in plotting:

• Choose integer values close to the vertex for \( x \), such as -1, 0, and 1.
• Calculate corresponding \( y \)-values: \( h(-1) = |-1| - 3 = -2 \), \( h(0) = |0| - 3 = -3 \), and \( h(1) = |1| - 3 = -2 \).

Connecting these points forms the V-shape, symmetric about the y-axis. Regularly plot additional points for further precision in larger graphs.

The vertex of a function is a crucial point, especially for absolute value functions and quadratics. For \( h(x) = |x| - 3 \), the vertex represents the lowest point in its V shape. With the transformation of the basic absolute value function \( y = |x| \), the vertex shifts directly downward by 3 units. This transformation moves the vertex from \( (0,0) \) to \( (0,-3) \).

• The vertex \((0, -3)\) is significant because it marks the minimum value of the function.
• All other values of \( h(x) \) are greater than \( -3 \).

By identifying the vertex first, graphing tasks become clearer as it provides an anchor for symmetry and direction. This makes it simpler to plot and verify additional points against the graph's main layout.