Graphs of Functions

 

Review: A function is a rule of correspondence that assigns to each element of a set A exactly one element of a set B.

A function consists of a set of numbers, called the domain, and a rule, given by the notation y = f(x), where each number in the domain determines one and only one number to form a set, called the range.

  1. The domain (set A above) of a function is the set of all possible inputs, that is, the largest set of real numbers for which y = f(x) makes sense. To indicate this, we use the phrase "the function must be defined over the domain."
  2. The range (set B above) of a function is the set of all possible outputs, that is, the largest set of all real numbers determined by applying the rule y = f(x) to the numbers in the domain.

Finding the Domain and Range using the Graph of a function

Below are three different methods of determining domain and range. They are algebraic, graphical and numerical (by table).

Algebraic Method

Example:

  • The Domain is the acceptable values you can input into the function for the x variable. Here, we solve the expression created by the square root symbol itself:

So, the domain is .

  • The Range is found by realizing the square root symbol produces numbers greater than or equal to zero, so the outputs, the range, is .

Example:

  • The Domain is found by realizing that you can substitute any real number in for the x variable, so the domain is .
  • The Range is found is found by realizing the largest value the output could be is y = 1. We are always subtracting positive results or zero from 1. So, the range is .

Example: (Find domain only.)

  • The Domain is found by setting the denominator equal to zero and solving.  This finds those x values that cause division by zero.
  • Those x values are then removed from the domain. So, the domain is all real numbers except -3, ie, .

 

Graphical Method

Note: Each graph below is on a standard window [-10, 10] in both x and y unless specified otherwise.

  • The domain is read as the extent of the graph in the direction of the x-axis, that is, it is read as the extent from left to right.
  • The range is read in as the extent of the graph the direction of the y-axis, that is, it is read the as the extent from down to

Example: Graph of

  • The graph extends to the left forever, and seems to end at x = 3. So, the domain is .
  • The graph lowest extent is y = 0, and it extends to up forever. So, the range is .

 

Example:Graph of

  • The graph extends left and right forever. So, the domain is .
  • The graph extends to the down forever, and seems to end at y = 1. So, the range is .

 

Numerical Method

  • The domain is read as the extent of the x values in the table feature.
  • The range is read as the extent of the y values in the table feature.
  • In both cases, some exploration of the table is necessary.

Example: Table of values for

         

  • The table has "ERROR" in the y column for x values greater than x = 3. So, the domain is
  • The table shows the smallest y value as zero. So, the range is.

 

Example: Table of values for

         

  • The table will show y values for any x value. So, the domain is .
  • If you scan the y values, it is not clear what the range is, but if you go to G-Solv and x-CAL, and enter y values greater than 1, you will see "Not Found" messages. So, the range is .

 

Vertical Line Test

The test for seeing if a graph is the graph of a function is to examine the graph and if any vertical line intersects the graph in more than one location, the graph is not that of a function. Otherwise it is the graph of a function.

 

Examples: Determine if the following graphs are graphs of functions.

Yes, since each domain value has only one range value associated with it.

No, since there exists at least one domain value with more than one range value associated with it.

 

 

 

Describe the Increasing and Decreasing Behaviors of a Function

Other information we are interested in when it comes to graphs is when they are increasing, decreasing, or neither.

  • When is a function increasing?
    • A function is increasing when it extends up as the graph extends to the right.
  • When is a function decreasing?
    • A function is decreasing when it extends down as the graph extends to the right.
  • When is a function neither increasing not decreasing?
    • A function is called constant when it is neither increasing nor decreasing.

How do we report this information?

Always report the information in terms as open intervals in terms of x values. This should make perfect sense since the x coordinate can never repeat in a function, but the y coordinate can repeat. So, to locate where a function is behaving, it is only useful to talk in terms of the input or x values.

Example: Describe the increasing, decreasing and constant behavior of the function whose graph is below.

Increasing: Decreasing:     Constant: Never!
Increasing:    Decreasing:    Constant:

 

 

Classify a function as Even or Odd

  • Even functions are symmetric with respect to the y-axis.

    Recall, that when we have symmetry with respect to the y-axis both (x, y) and (-x, y) are on the graphs. Then.

  • Odd functions are symmetric with respect to the origin.

    Recall, that when we have symmetry with respect to the origin both (x, y) and (-x, -y) are on the graphs. Then .

  • If a graph of a function has neither type of symmetry, then it is neither even nor odd.

 

Examples: Determine whether each function is even, odd, or neither.

Algebraic Method

Example:

  • Substitute -x for x everywhere in the rule for f(x):

  • Analyze the result.
    1. First, because there is a sign change. So, there no symmetry with respect to the y-axis and the function is not even.
    2. Also, because not all signs change. So, there no symmetry with respect to the origin and the function is not odd.
    3. So  f   is neither even or odd.

Example:

  • Analyze the result.
    1. First, because there is a sign change. So, there no symmetry with respect to the y-axis and the function is not even.
    2. But, because all signs change. So, we have symmetry with respect to the origin and the function is odd.

Example:

  • Analyze the result.
    1. First, because there is no sign change. So, there is symmetry with respect to the y-axis and the function is even.
    2. But, because all signs don't change. So, we have no symmetry with respect to the origin and the function is not odd.

 

Graphical Method then Numerical Method

Example:

The graph appears to have the symmetry about the y axis:  

Confirmation is found by examining the table:  

  • Observe the pattern: If (x, y) is on the graph, then (-x, y) is on the graph too.
  • Observe the pairs of points: (2, 4) and (-2, 4) & (3, 6) and (-3, 6).
  • The function is even.

Example:

The graph appears to have the symmetry about the origin:   

Confirmation is found by examining the table

  • Observe the pattern: If (x, y) is on the graph, then (-x,-y) is on the graph too.
  • Observe the pairs of points:(1, -2)and (-1, 2) & (2, -4) and (-2, 4).
  • The function is odd.

 

Identify the Six Common Graphs

Become familiar with the following six common graphs. In the next section, we will shift and transform the graphs of these functions. Graph each function on your calculator, then adjust the scalings.

 

Practice Problems

 

Practice One:

Find the domain of the function.

    5. Why are the domains of the last two different?

Practice Two:

An open box is to be made from a square piece of material 14 inches on a side by cutting equal squares of length x from the corners and turning up the sides.

    1. Write the volume V as a function of x.
    2. What is the domain of this function?
 

Practice Three:

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 100 inches.

    1. Write the volume as a function of x.
    2. What is the domain of the function?

 

Practice Four:

From 1990 to 1996, the total number of radio stations that operated with a country format can be approximated by the function , where R(t) is the number of radio stations and t = 0 represents 1990.

    1. Graph the function.
    2. Estimate the years in which the number of country stations was increasing and the years in which the number of country stations was decreasing.
    3. Estimate the maximum number of country stations from 1990 to 1996.
    4. Estimate the minimum number of country stations from 1990 to 1996.

 

© 2003 Jay Abramson, Terry Turner and the ASU School of Mathematical and Statistical Sciences. - All rights reserved.