Properties of Function

Introduction

Graphing functions is not that different from graphing equations. The biggest difference is that for each value of x in the domain we have only one y-value. In addition to graphing functions, we will discuss being able to identify whether or not a graph is the graph of a function. We will also discuss symmetries in the graph of a function.

Vertical Line Test

Given a set of points in the xy-plane, the points form the graph of a function if and only if the set of points intersects any vertical line in at most one point. This is called the vertical line test.

Non-example: A circle fails to be the graph of a function since the vertical diameter intersects the graph of the circle in two points.

Symmetry

Just as we have discussed the symmetry in the graph of an equation, we can discuss the symmetry in the graph of a function.

We call a function even if its graph is symmetric across the y-axis. In terms of the function definition, this means f(-x) = f(x).

We call a function odd if its graph is symmetric with respect to the origin. Another way to express this to to say that f(-x) = -f(x).

Some basic examples of even functions are $x^{2},\,x^{4}$, while some examples of odd functions are $x,x^{3}$

Increasing, Decreasing or Constant

We say a function is increasing(decreasing) if the value of the function is increasing(decreasing) as the x-value increases. A function is constant if it is neither increasing nor decreasing.

Using these definitions we can talk about the intervals over which a funciton is increasing, decreasing, or constant.

A more formal way to express these three concepts is as follows:

1) A function is increasing on an open interval I if, $f(x_{1})>f(x_{2})$ for any $x_{1}>x_{2},{\text{ and }}x_{1},x_{2}{\text{ in I.}}$

2) A function is decrasing on an open interval I if, $f(x_{1})<f(x_{2})$ for any $x_{1}<x_{2},{\text{ and }}x_{1},x_{2}{\text{ in I.}}$

3) A function is constant on an open interval I if, $f(x_{1})=f(x_{2})$ for any $x_{1}\neq x_{2},{\text{ and }}x_{1},x_{2}{\text{ in I.}}$

Maxima and Minima

Now that we can identify intervals where the function is increasing and decreasing, we can discuss local maxima/minima, which are points where the function switches from increasing to decreasing, or vice versa. For an example you can think of the apex (highest point) a tossed ball reaches. Once the ball reaches its apex it stops going higher and starts falling.

We now follow with a more formal definition:

A function f has a local maxima(minima) at a point c if there is an open interval containing c such that $f(x)\leq f(c),f(x)\geq f(c){\text{ respectively.}}$ In that case we call f(c) a local maxima(minima) value of f.

If a local maxima(minima) is larger(smaller) that the function value at other point, then we call the value the absolute maximum(minimum).

This leads us to the Extreme Value Theorem:

If a f is a continuous function on a closed interval [a, b], then f has an absolute maximum and an absolute minimum on [a, b]

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