Therefore the domain of any exponential function consists of all real numbers ( − ∞, ∞ ). Using rational exponents in this manner, an approximation of 2 7 can be obtained to any level of accuracy. Consider 2 7, where the exponent is an irrational number in the range, Up to this point, rational exponents have been defined but irrational exponents have not. For example, the initial population of a town is 10,000, and the equation is defined as: The population of the town after five years will be 15,000, and this is a linear growth relationship. Here we can see the exponent is the variable. Linear growth graphs are considered to be an upward-sloping straight line. has the form,į ( x ) = b x E x p o n e n t i a l F u n c t i o nįor example, if the base b is equal to 2, then we have the exponential function defined by f ( x ) = 2 x. Given a real number b > 0 where b ≠ 1 an exponential function Any function with a definition of the form f ( x ) = b x where b > 0 and b ≠ 1. In this section we explore functions with a constant base and variable exponents. We have studied functions with variable bases and constant exponents such as x 2 or y − 3. In this section well examine graphs that show exponential growth and.
At this point in our study of algebra we begin to look at transcendental functions or functions that seem to “transcend” algebra. As with other functions, ordered pairs can be used to graph an exponential function.
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