# 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 ${\displaystyle x^{2},\,x^{4}}$, while some examples of odd functions are ${\displaystyle x,x^{3}}$

## 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, ${\displaystyle f(x_{1})>f(x_{2})}$ for any ${\displaystyle x_{1}>x_{2},{\text{ and }}x_{1},x_{2}{\text{ in I.}}}$

2) A function is decrasing on an open interval I if, ${\displaystyle f(x_{1}) for any ${\displaystyle x_{1}

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

## 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 ${\displaystyle f(x)\leq f(c),f(x)\geq f(c){\text{ respectively.}}}$ 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]