Power Functions

Algebraic Representation

f(x) = a x b

It is easy to confuse power functions with exponential functions. Both have a basic form that is given by two parameters. Both forms look very similar. In exponential functions, a fixed base is raised to a variable exponent. In power functions, however, a variable base is raised to a fixed exponent.

The parameter  a  serves as a simple scaling factor, moving the values of  xb  up or down as  a  increases or decreases, respectively

The parameter  b , called either the exponent or the power, determines the function's rates of growth or decay. Depending on whether it is positive or negative, a whole number or a fraction,  b  will also determine the function's overall shape and behavior.

More so than other simple families like lines, exponentials, and logs, members of the power family can exhibit many distinctive behaviors.

Notice that when  b = 0  the function simplifies to  f(x) = ax0 = a1 = a , a constant function with an output of  a  for every input. When  b > 0 ,  f(0) = a0b = 0 . That is, every power function with a positive exponent passes through  (0, 0) . When  b < 0 ,  f(0)  is undefined. (Remember,  x–b = 1 / xb .) Thus, power functions with negative exponents have no y-intercepts. Power functions with negative, whole number exponents like  x–1  or  x–2  are simple examples of rational functions, and for these functions  x = 0  is an example of a singularity.

To appreciate the variety of behaviors among members of the power family, consider two simple cases:

The difference between odd and even powers only hints at the differences among power functions.

Another helpful distinction separates functions with whole number (integer) powers from those with fractional powers. (We leave the consideration of irrational powers to calculus.)

It is difficult to single out one key    algebraic property of power functions from all of this variety. Perhaps it is best to say simply that powers behave like powers, with an appreciation for the many different behaviors that this can encompass.

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