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 Dependent Variable

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# Exponential Functions

## Laws of Exponents

If s, t, a and b are real numbers, with a > 0 and b > 0, then:

1) as Â· at = as + t

2) (as)t = ast

3) (ab)s = as Â· bs

4)

5)

6)

7) 1s = 0

8) a0 = 1

## Exponential Functions

Definition: If b is a positive real number, b 1, the exponential function with base b is f(x) = bx , for every real number x.

Examples:

 f(x) = 2x f(x) = 3x

## Properties of Exponential Functions of the Form f(x) = bx

1. bx > 0 for all numbers, x; so, the range of these functions is (0, ).

2. Their domains will be (-,).

3. All will have y-intercepts at (0,1) and no x-intercepts.

4. The x-axis is a horizontal asymptote.

5. If b > 1, f(x) = bx is an increasing function. If 0 < b < 1, f(x) = bx is a decreasing function.

6. They are all one-to-one.

Here is a graph with f(x) = 2x , f(x) = 3x and f(x) = 4x all on the same screen. Can you tell which is which?

## The Natural Exponential Function: f(x) = ex

e is a special number in mathematics, and is approximately 2.7182818. Since e > 1, the graph of the exponential function is shaped like the graph of f(x) = 2x. It is an increasing function. ex > 0 for all numbers x. Here is the graph of f(x) = ex:

 Exponential Growth FunctionsAny function of the form y = a(bx), b > 1, a > 0, is an exponential growth function. Its graph will increase on its entire domain. Exponential Decay FunctionAny function of the form y = a(bx), b > 1, a > 0, or y = a(c ), 0 c 1, a 0, is an exponential decay function, and will decrease on its domain.