Definition

A power function is a function of the form,

f(x) = ax^p,

where a ≠ 0 is a constant and p is a real number. Some examples of power functions include:

Root functions, such as are examples of power functions. Graphically, power functions can resemble exponential or logarithmic functions for some values of x. However, as x gets very large, power functions and exponential or logarithmic functions begin to diverge from one another. An exponentially growing function will overtake a growing power function for large values of x. On the other hand, growing power functions will overtake logarithmic functions for large values of x.

Domain and Range

The domain of a power function depends on the value of the power p. We will look at each case separately.

1. p is a non-negative integer

The domain is all real numbers (i.e. (− ∞,∞)).

2. p is a negative integer

The domain is all real numbers not including zero (i.e. (−∞, 0) ∪ (0,∞) or {x|x ≠ 0}). We will revisit this case when we study rational functions.

3. p is a rational number expressed in lowest terms as r / s and s is even

A. p > 0

The domain is non-negative real numbers (i.e. [0,∞) or {x|x ≥ 0}).

B. p < 0

The domain is positive real numbers (i.e. (0,∞) or {x|x > 0}).

4. p is a rational number expressed in lowest terms as r / s and s is odd

A. p > 0

The domain is all real numbers.

B. p < 0

The domain is all real numbers not including zero.

5. p is an irrational number

A. p > 0

The domain is all non-negative real numbers.

B. p < 0

The domain is all positive real numbers.

 

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In the next section we will study the graphs of power functions.

Graphing power functions