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Exponential functions are mathematical expressions where the variable appears in the exponent. In the case of the inequality y > 2^{x}, the base is 2 and the exponent is x. This kind of function is known for its rapid growth rate when x is positive, and its rapid decay towards zero when x is negative.

Understanding exponential functions is crucial when solving inequalities because these functions do not grow or decrease at a constant rate. For instance, 2^{x} will double with each increase in x by 1 (for positive x), which creates a distinct, J-shaped curve when graphed. Moreover, the function will never reach zero, it just gets infinitely closer, which explains why the graphed curve gets closer to the x-axis but never touches it as x goes to negative infinity.

Inequalities, such as y > 2^{x}, are relational statements indicating that one quantity is greater than (or less than, depending on the inequality) another. In this context, we are focused on finding all the y values that are greater than 2^{x} for any given x.

Distinguishing between different types of inequalities is vital. If the inequality were y ≥ 2^{x} instead, it would include all the points where y is exactly equal to 2^{x}, and the curve on the graph would be solid. With a strict inequality (y > 2^{x}), all points exactly on the curve y = 2^{x} are not included in the solution. This is often visually indicated on graphs by dashing the line. The solutions to this inequality live in the region above the exponential curve, which shows the dynamic between the exponential function and inequality solving.

Graphing is a powerful tool for visualizing the relationship described by mathematical equations and inequalities. When graphing an exponential inequality like y > 2^{x}, the first step is to graph the related exponential function y = 2^{x}. Knowing key characteristics of exponential graphs helps: they pass through (0,1), as any number to the power of zero is one, and they rise quickly as x increases when the base is greater than one.

After plotting the curve of the exponential function, the next step in graphing inequalities is to show which side of the curve is relevant to the inequality. This involves shading the appropriate region. Since y must be greater than 2^{x}, the region above the curve is shaded. The curve itself is dashed to indicate that points on the curve do not satisfy the inequality. By incorporating these graphing techniques, students can easily visualize the solution set and understand the relation between the function and the inequality.