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# Basic Exponential Functions: f(x) = bx by Christopher Monahan

The parent function for the exponential function is y = bx, where b is a positive number not equal to 1. One is not included because 1 raised to any power is still 1, and the graph of this function is a horizontal line.

What are the domain and range of the exponential function? Because b is positive, the exponent can be any real number, so the domain is the set of reals. A positive number raised to any power will yield a positive answer. (Recall that a negative exponent indicates a reciprocal, and the reciprocal of a positive number is a positive number. A fractional exponent indicates a radical, and the root of a positive number is a positive number.) Therefore, the range of an exponential function is y > 0. All exponential functions of the form y = b x pass through the point (0, 1).

The graph of an exponential function for values of b > 1 increases as the values of x increase. For example, look at the function f(x) = 2x, its graph, and a table of values for this function.

As the values of x increase, the values of y grow continuously. As the value of x goes toward negative infinity, the value of y gets very close to 0. In fact, the line y = 0 becomes a boundary for the graph. The graph gets very close to this line but will not touch or cross it as these negative values of x become infinitely large in magnitude. This line is called a horizontal asymptote.

The rate at which the exponential graph will grow depends on the value of the base. For values larger than 2, the graph will grow that much more quickly; for values between 1 and 2, the graph will grow less quickly.

What happens if the value of b is between 0 and 1? Consider the function

. When you examine the graph and its table of values, you see that g(x) is a reflection of f(x) across the y-axis.

The values of y decrease as the values of x increase, and the graph has a horizontal asymptote at y = 0.