Exponential functions (§6.3)

Recall that a power function is a function f of the form

f(x) = xⁿ,

for some constant n called the exponent of the function. 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.

Recall that a linear function is a function f of the form

f(x) = mx + b,

for some constants m and b. Analogously, a generalised exponential function is a function f of the form

f(x) = Cb^x,

for some constants b and C.

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

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

The domain of a generalised 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 generalised exponential function is the set of all real numbers with the same sign as C. If C > 0 and b > 1, then the generalised exponential function is increasing; if either of these is reversed, then the function is decreasing; if both are reversed, then it's increasing again. (If b = 1 or C = 0, then the exponential 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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