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Exponential Functions

Exponents can be variables. Variable exponents obey all the properties of exponents listed in Properties of Exponents.

An exponential function is a function that contains a variable exponent. For example, f (x) = 2x and g(x) = 5ƒ3x are exponential functions. We can graph exponential functions. Here is the graph of f (x) = 2x:

Figure %: f (x) = 2x
The graph has a horizontal asymptote at y = 0, because 2x > 0 for all x. It passes through the point (0, 1).

We can translate this graph. For example, we can shift the graph down 3 units and left 5 units. Here is the graph of f (x) = 2x+5 - 3:

Figure %: f (x) = 2x+5 - 3
This graph has a horizontal asymptote at y = - 3 and passes through the point (- 5, - 2).

We can stretch and shrink the graph vertically by multiplying the output by a constant--see Stretches. For example, f (x) = 3ƒ2x is stretched vertically by a factor of 3:

Figure %: f (x) = 3ƒ2x
This graph has a horizontal asymptote at y = 0 and passes through the point (0, 3).

We can also graph exponential functions with other bases, such as f (x) = 3x and f (x) = 4x. We can think of these graphs as differing from the graph of f (x) = 2x by a horizontal stretch or shrink: when we multiply the input of f (x) = 2x by 2, we get f (x) = 22x = (22)x = 4x. Thus, the graph of f (x) = 4x is shrunk horizontally by a factor of 2 from f (x) = 2x:

Figure %: f (x) = 4x
This graph has a horizontal asymptote at y = 0 and passes through the point (0, 1).

The graph of f (x) = ax does not always differ from f (x) = 2x by a rational factor. Thus, it is useful to think of each base individually, and to think of a different base as a horizontal stretch for comparison purposes only.

The graph of an exponential function can also be reflected over the x-axis or the y-axis, and rotated around the origin, as in Heading .

The general form of an exponential function is f (x) = cƒax-h + k, where a is a positive constant and a≠1. a is called the base. The graph has a horizontal asymptote of y = k and passes through the point (h, c + k).

The domain of f (x) is and the range of f (x) is .

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