PreCalculus

10.4. Classifying functions

The behavior of a function may be described in several useful ways.

Increasing and decreasing functions

If the value returned by a function increases as the value of the independent variable increases throughout its domain, then it is an increasing function. That is, the following is always true:

f(x₁) < f(x₂) for all x₁ < x₂

Conversely, if the value returned by a function decreases as the value of the independent variable increases, then it is a decreasing function. That is, the following is always true:

f(x₁) > f(x₂) for all x₁ < x₂

For a constant function, the following is true:

f(x₁) = f(x₂) for all x₁, x₂

A function may be increasing, decreasing, or constant in portions of its domain. For example, consider the following function:

f: R -> R, f(x) = x²

It decreases in the interval (-∞, 0], and increases in the interval [0, +∞).

Even and odd functions

We say that a function is even if it satisfies the equation:

f(-x) = f(x)

for all values of x. All even power functions:

f(x) = x²ⁿ

where n is an integer satisfy this equation. The graph of an even function is symmetric about the y-axis.

A function is odd if it satisfies the equation:

f(-x) = -f(x)

for all values of x. All odd power functions:

f(x) = x²ⁿ⁺¹

where n is an integer satisfy this equation. The graph of an odd function is symmetric about the origin.

Periodic functions

A function is periodic if it repeats itself consistently with a specific period. The period of a function is the length of the shortest interval over which it repeats. That is, a periodic function satisfies the equation:

f(x + p) = f(x)

for all inputs x such that x + p and x are both in the domain of the function. The period is the smallest value of p where the above equation is satisfied over the domain of the function.

These trigonometric functions repeat with period 2π

f(x) = sin x
f(x) = cos x

Piecewise functions

A piecewise function is also known as a compound function. A function may be defined piecewise with a rule that gives more than one formula for different intervals in the domain of the function. For example, the mathematical definition of the sign function is.

          +1     x > 0
sgn(x) =   0     x = 0
          -1     x < 0

This function would be graphed as follows to show how it behaves where the jumps in value occur.

Figure 10-3: Graph of sgn(x)

Note that the open end of an interval is drawn as an open circle. The closed end of an interval would be drawn as a filled in circle.