Power Functions


Significance

The functions y = xn are power functions, so polynomials are made from power functions. More general exponents arise naturally in applications. For example, volume increases as the (3/2)th power of the surface area. "Root" functions such as the sqrt(x) and cube root of x are power functions.


Standard Notation

Power functions are functions of the form f(x) = x2, f(x) = 4 x-3, f(x) = -9 x1/2, and so on. The general form is f(x)= kxp where p is any real number and k is non-zero.


Rules of exponentiation

The Rules of exponentiation are:
  • x(p + q) = (xp)(xq).

  • xpq = (xp)q.

  • x -p = 1/xp.

  • x(1/p) is the p-th root of x.

  • x0 = 1 for any xneq0. 00 is undefined.

  • (xy)p = (xp)(yp).
In particular, the root functions such f(x) = sqrt (x) are really just special cases of power functions with, for example, sqrt (x) = x(1/2).


Domains of Power Functions

If p is a non-zero integer, then the domain of the power function f(x) = kxp consists of all real numbers. For rational exponents p, xp is always defined for positive x, but we cannot extract an even root of a negative number. Thus x(1/4) is not defined for any negative real numbers. Neither is x(3/4) (the fourth root of x cubed).

Any rational number p can be written in the form p = r/s where all common factors of r and s have been cancelled. When this has been done, kxp has domain

  • All real numbers if s is odd
  • All non-negative real numbers if s is even.
If p is a real number which is not rational (called an irrational number), then the domain of xp consists of all non-negative real numbers.


Graphs of Power Functions

The shape of the graph of y = kxp depends on the sign of k and where p lies. The effect of the coefficient k is to scale and, if negative, flip the graph about the x-axis. All graphs of y = xp pass through the point (1,1). Beyond that, the shape and growth for large positive x is given by:

p > 1 Concave up.
Grows as x grows large.

y = x2 (RED), y = x (GREEN)

y = x0 (YELLOW)

y = x1/2 (BLUE), y = x -2 (BLACK)

p = 1 The straight line
y = x.
0 <p< 1 Concave down
Grows as x grows large.
p = 0 The straight line
y = 1.
p < 0 Concave up
Approaches 0 as x grows large


Calculations with Power Functions

The typical equation involving a power function is an equation of the form

kxp = b.

The solution to such an equation, if p solution exists, is given by

x = (b/k)(1/p).

as can be easily checked using the rules for exponentiation.

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