Exponential functions (§5.3)

The next couple of weeks will be about exponential and logarithmic functions. Logarithms are particularly useful in many applications of mathematics. But first, I'll introduce exponential functions using a couple of analogies with other kinds of functions.

A power function is a function f of the form

f(x) = xⁿ,

for some constant n called the exponent of the function; we've seen lots of examples of power functions up to now. In contrast, an exponential function is a function f of the form

f(x) = b^x,

for some constant b called the base of the function. The base should be a positive number, so that b^x makes sense for every real number x.

Now recall that a linear function is a function f of the form

f(x) = mx + b,

for some constants m (the rate of change) and b (the initial value). Analogously, a generalized exponential function is a function f of the form

f(x) = Cb^x,

for some constants b (the base) and C (the initial value).

If you don't remember any other values of a generalized exponential function, remember these:

f(0) = Cb⁰ = C,
f(1) = Cb¹ = C · b,
f(−1) = Cb⁻¹ = C/b.

Just as you can find the rate of change of a linear function by subtracting f(x + 1) − f(x), so you can find the base of a generalized exponential function by dividing f(x + 1)/f(x).

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, b^x 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 e^x from x; in particular, you can calculate e itself, as e¹, using this button.

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