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 x^1/2, and so on. The general form is f(x)= kx^p where p is any real number and k is non-zero.

Rules of exponentiation

The Rules of exponentiation are:

x^{(p + q)} = (x^p)(x^q).
x^pq = (x^p)^q.
x^-p = 1/x^p.
x^(1/p) is the p-th root of x.
x⁰ = 1 for any x0. 0⁰ is undefined.
(xy)^p = (x^p)(y^p).

In particular, the root functions such f(x) = sqrt (x) are really just special cases of power functions with, for example, sqrt (x) = x.

Domains of Power Functions

If p is a non-zero integer, then the domain of the power function f(x) = kx^p consists of all real numbers. For rational exponents p, x^p 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, kx^p 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 x^p consists of all non-negative real numbers.

Graphs of Power Functions

The shape of the graph of y = kx^p 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 = x^p 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 = x^1/2 (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

kx^p = 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.