Unit 1 Functions and their Graphs 1.4-1.7

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Unit 1-Functions and Their Graphs

We continue our focus on the definition of a function, how to graph a function, and the relationship of functions and their graphs as we learn to analyze a function in the next section.
Follow this link for a list of Graphing Calculator Skills needed for this chapter.

Section 1.4 Graphs of Functions
Main Points and Objectives

Domain and Range.
You should be able to identify the domain and the range from a graph of the function, and for many functions be able to identify the domain and the range from analyzing where the function exits agebraically. The notation used for intervals can be confusing. When discussing domain and range, (a,b) means the interval from a to b but not including the endpoints a or b. This is the same notation use for coordinate points and can be confusing to many students. Please be careful. When the endpoints are included bracket notation is used. [-2,6) means the interval containing all the points from -2 to 6, including 2 but not including 6.

The domain of this relation is [-3,3].
The range is[-2, 2].

The domain of this function is [-4,4].

The range is [-1, 1].

If it is assumed the the function goes on and on in both directions forever, the domain of the above function becomes (-oo, oo) Note: oo means infinity.

Click on the icon to explore domain graphically

QUIZ for Domain and Range.
In order to view this quiz and other material later in the course, you will need to download a free version of Adobe Acrobat Reader. Please click here to get the program.

Quiz 1
1. E-mail, just the solutions and explanations in text. OR
2. Fax the quiz back to me.The fax number is posted on the WebBoard

Vertical Line Test.

This method is used to determine whether a relation is indeed a function. Please refer to page 117 for more examples.
Since a vertical line passes through more than one point, this relation is not a function.

Increasing and Decreasing Functions

When graphing functions on the calculator, it is easy to see if the function is increasing, decreasing, or even remaining constant. The table feature of the calculator can also be used for this. Note: The interval over which a function is increasing or decreasing is the x values.
This function appears to be increasing on (-oo,-2) U (1, oo) and decreasing on (-2, 1).

Relative maximum and minimum

These are the points at which a function changes from increasing to decreasing or from decreasing to increasing. They can be approximated with the TRACE feature of the calculator, or using the table feature, and also be found using the CALC menu.
The relative maximum appears to be approximately the point (-2,8) , and the relative minimum appears to be approximately the point (1, -4)
TRY it now!! Use your graphing calculator with a Standard window to find the relative maximum of :
y = -2x³ + 3x -3

Step function

This is called the greatest integer function and is very useful in real-world applications. The calculator has a feature which can help graph these functions, INT . This is found in the MATH, NUM menu. Please email for more information or check with the video.

Example of a "step function".
y1 = 2int (.5x)
graphed in "dot mode" .

Section 1.5 Shifting, Reflecting, and Stretching Graphs
This section explores the relation between functions and how their graphs are shifted vertically or horizontally, and stretched. This topic will be explored in further detail as the course progresses but you can explore now with the LiveMath Notebook below.

Click on the icon to explore the online function grapher .

Section 1.6 Combinations of Functions

Sum, Difference, Product, and Quotient of Functions.
The domain of the combination function is the domain which is common to both.

Composition of Functions.

The graphing calculator can do a numerical composition by using the cut and paste of the y1 and y2 functions found in the Y-VARS menu. Refer to the manual for this or email me for help if you cannot get to these functions on the calculator.

Click on the LiveMath image to explore functions and compositions.

Section 1.7 Inverse Functions

The inverse of a function

is a function such that f(f ^-1(x)) = x and (f ^-1(f(x)) = x for all x in the domain.. This topic is explored in great detail in the text and on the CD. Please read the blue section on page 155 to learn to use the DRAW feature of the calculator to draw the inverse of a function. In this manner you can see what it looks like before determining it algebraically, and also to check your work. You cannot trace a function with DRAW, but you can see that for every coordinate, (x,y) of the function, (-x, -y) is a point on the inverse. The inverse of a function is not always a function itself . Check the vertical line test for that.
The is the graph of y = x³ and its inverse.
To explore inverse functions with LiveMath, click the icon.

One to One. The horizontal line test is used to check for one-to-oneness.
This function is not one-to-one as it does not pass the horizontal line test. It has no inverse.

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