## The Biology Project > Biomath > Power Function > Basics

## Power Functions

Basics

 Definition A power function is a function of the form, f(x) = axp, 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.   ***** In the next section we will study the graphs of power functions. Graphing power functions

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

April 2006
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