Basic Exponential Functions: f(x) = bx
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.
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