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What Are Power and Root Functions?

A power function is a special case of a polynomial function. A power function consists of only one term—a polynomial of the form axn.

Theory

Power and Root Functions

A power function is a function where f(x) is given as a number multiplied by an arbitrary power of x. The function can be written as

f(x) = a xn

If n is a fraction, we call the power function a root function, since it can be rewritten using the formula

xm n = xmn

Below is a brief description of how each function behaves for different values of n.

  • If n is a even number, then you get a parabola.

    Graph of a power function where n is an even number

  • If n is an odd number, you get graphs that are extended along the entire y-axis.

    Graph of a power function where n is an odd number

  • If n = 0, you get a straight line that intersects y = a.

    Graph of a power function where n = 0

  • If n < 0, you get rational functions.

    Graph of a power function where n < 0

  • If n (n is a fraction), you get a root function.

    Graph of a power function where n is a fraction

  • If n is in the form n = k 2m, and if k and 2m have no factors in common, then the graph begins in the origin.

    Graph of a power function where n is a reduced fraction

Note! Root functions are defined only for positive values of x, since you can only take the even root (x, x4, x6,) of numbers greater than or equal to 0.

Example 1

f(x) = 5x1 2 = 5x

is a power function and a root equation

Example 2

f(x) = 2x4

is a power function and a polynomial function.

Example 3

f(x) = 4x2 = 4 x2

is a power function and a rational function.

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