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An exponential function is a function of the form f(x) = a * b^x, where a is a constant, b > 0, and b ≠ 1. Exponential functions exhibit a constant rate of growth or decay depending on the value of the constant b. If b > 1, the function shows exponential growth, and if 0 < b < 1, the function shows exponential decay.

When choosing values for the variable x to create a table of values for an exponential function, it is important to consider the following characteristics of the function: 1. Exponential growth: As x increases in value, the value of the function, f(x), will increase rapidly. To capture this growth in the table, the x-values should be chosen at "regular" intervals, such as integer values or equally spaced increments. 2. Exponential decay: As x decreases in value, the value of the function, f(x), decreases rapidly. To capture this decay in the table, x-values should also be chosen at regular intervals, focusing on negative integers or equally spaced decrements. 3. The constant a: The constant a influences the overall shape of the function. If a > 0, the function will be increasing, and if a < 0, the function will be decreasing. 4. The constant b: The constant b determines the rate of growth or decay of the function.

Based on the characteristics of exponential functions discussed above, appropriate values for the variable x can be chosen as follows: 1. Choose an interval for x-values, which covers the domain of interest (e.g., integer values, fractions, or decimals). 2. Consider including some negative values for x, since they help show how the function behaves as x decreases. 3. Choose a few positive values for x to see how the function behaves as x increases. 4. If necessary, consider choosing more points around areas where the function is changing more rapidly to provide a more accurate representation of the function. In summary, when making a table of values to graph an exponential function, choose x-values at regular intervals, including both negative and positive values, to provide a clear representation of the function's behavior, capturing both its growth and decay aspects.