E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 2.4 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

2.4


The Graph of a Functions

Domain of a Function Vertical Line Test Piecewise Function



Graphing a function

The graph of a function f is the set of all point (x, y) in the xy-plane with x in the domain of the function and .



Example 1Specify the domains and ranges of the functions whose graph appears in the following figures:



Example 1 Solution

The domain of a given graph is the part of the x-axis used, while the range is the part of the y-axis used in the graph. Therefore, the domains and ranges of the functions , are as given in the following table.

Graph Function Domain Range



Example 2Find domain, range and intercepts of . Also, examine the symmetry, and sketch graph of

Domain: Range: x-intercepts: and y-intercept:

We also notice that the graph is symmetric about the y-axis because the equation does not change when is replaced by . To sketch the graph we obtain a few more points and we sketch the graph



Example 3Sketch the graph of the function

ymust satisfy y

must satisf

We observe that for and , the function is equal to the function . Therefore, has the same graph as for and . For , , which means that

the graph of on the interval will be a reflection of the graph of about the x-axis. Thus, the graph of is given by:



Vertical Line Test

It is important to understand that not every graph represents a function. For example, in the given graph the vertical, line L

A graph in the xy-plane represents a function , provided that every vertical line intersects the graph at most at one point.

Vertical Line Test

meets the graph twice. This means that there are two y-values for the same x where the line L meets the x-axis. Therefore, does not represent a function.



Example 4Determine which of graphs represent the graph of a function.

While, the graphs in figures (b), (d), (e) and (f) do belong to functions.

Using the vertical line test, we see that the graphs in figures (a) and (c) are not graphs for functions.



Graph of Basic Functions

Below are graphs for six basic functions together with some important information about the functions.



Graph of Basic Functions…cont….



Graph of Basic Functions…cont….



Piecewise Functions

All functions considered earlier are defined by one single equation. In some cases, however, functions may be defined by combining several equations. For example,

represent cases like this and is called a piecewise defined function.



Example 6Find , , and if

• We substituted in the expression because is less than .

• We substituted in the expression because lies in the interval , thus .

• We substituted and in the expression because and both are greater than or equal to . Thus, and



Example 7Sketch graph of

For the graph of , we use the squaring function for x less than 2, horizontal line for with , and the reciprocal function for . Then the following is the graph for .



Increasing, Decreasing and Constant Functions

A function is said to be increasing (decreasing) on an interval within the domain of if whenever are two numbers inside this interval we have

A function is said to be constant on an interval if

for every in this interval, we have .

Observe that the graph of a constant is a horizontal line.



Turning Points of a Graph

A point in the domain of is called a turning point if the graph of changes from increasing to decreasing at or vise versa.

A turning point is called a maximum point (minimum point) if the graph of at changes from increasing to decreasing (decreasing to increasing).



Example 8Determine the interval(s) where the given graph of is constant, increasing or decreasing and find maximum and minimum points.

is decreasing on The turning point is a maximum, while the turning point is a minimum

is constant on

is increasing on

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 2.4 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals