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In mathematics, the domain and range of a graph are foundational concepts to understand. These terms describe the x-values and y-values over which a graph extends.Domain of a Graph
The domain refers to all possible input values (x-values) for which the graph is defined. If we imagine a graph on a sheet of graph paper, the domain represents the horizontal stretch of the graph.
For our specific case, the domain includes all x-values between \( -3 \) and \( 5 \). This means every point plotted on our graph must respect these limits. Think of it as a restriction telling us that our graph cannot extend beyond \( -3 \) on the left and \( 5 \) on the right.
Range of a Graph
The range, on the other hand, refers to all possible output values (y-values) that a graph can take. It represents the vertical stretch of a graph.
For our exercise, the range is from \( -4 \) to \( 4 \). Thus, all y-values of the points on our graph should be captured within this interval, ensuring our graph doesn't dip below \( -4 \) at the bottom or rise above \( 4 \) at the top. Understanding and marking these boundaries is crucial before plotting more elements on a graph.

Not all graphs depict functions. A graph is deemed a function if every x-value corresponds to exactly one y-value. However, in some cases, a graph might fail this condition – leading to what's known as a non-function graph.Characteristics of Non-Function Graphs
To create a non-function graph, you can introduce elements, such as vertical line segments or closed curves, like circles. This results in some x-values having multiple y-values.
In our task, simply plotting the points \( (-2,3) \) and \( (3,-2) \) isn't enough. We must introduce additional structure, like a vertical line or closed curve, to ensure that certain x-values meet multiple y-values. This creates a situation where one input (x-value) reaches out to several outputs (y-values), disqualifying our graph from being a function.
Visual creativity is key to ensuring the graph remains effective within the given domain and range while aligning with this non-function criterion.

The vertical line test is a visual way to determine if a graph represents a function. This simple test involves understanding how a graph behaves with respect to vertical lines. How Does the Vertical Line Test Work?
Imagine drawing vertical lines along various x-values across your graph. If any of these vertical lines intersect the graph at more than one point, the graph doesn't depict a function.
For our exercise, we need a graph that fails this vertical line test. By incorporating structures like vertical line segments or closed curves within our domain and range, we ensure that some x-values correspond to multiple y-values. Thus, some vertical lines will hit the graph in more than one place, confirming it's a non-function.
In summary, while plotting, one must ensure that the graph has a breadth that allows vertical line crossings at multiple points to achieve a non-functional status effectively.