A power function is a function f of the form
Now recall that a linear function is a function f of the form
If you don't remember any other values of a generalized exponential function, remember these:
The domain of a generalized exponential function is the set of all real numbers; as long as b ≠ 1, the range of the exponential function with that base is the set of all positive numbers. (Because b is positive, bx is also positive.) Then the range of a generalized exponential function is the set of all real numbers with the same sign as C. (But if b = 1 or C = 0, then the range consists of only C.) If C > 0 and b > 1, then the generalized exponential function is increasing; if either of these is reversed, then the function is decreasing; if both are reversed, then it's increasing again. (But if b = 1 or C = 0, then the function is constant.)
Besides numbers such as 10, 2, and 1/2, which you are familiar with, another common choice of base is a special number, about 2.72, known as e. The importance of this number e will become clear when we look at applications. Many calculators have a button that calculates ex from x; in particular, you can calculate e itself, as e1, using this button.
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