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 nonnegative 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 {xx ≠ 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 nonnegative real numbers (i.e. [0,∞) or {xx ≥ 0}).
B. p < 0
The domain is positive real numbers (i.e. (0,∞) or {xx > 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 nonnegative 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
