Lesson Functions, Domain, and Range in Living Color

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About rapaljer: Retired Professor of Mathematics from Seminole Community College after 36 years.

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2.07  Functions, Domain, and Range

          College Algebra: One Step at a Time,  Pages  265 - 273:  #53, 70, 75        

                                                                                                             Pages  274 - 280:  #2, 18, 25, 26, 35, 38, 44, 46

                                                                                             Pages  283 - 288:  #3, 4, 18, 19, 25       

                                                          Extra Problem        

 

Dr. Robert J. Rapalje

Seminole Community College

Sanford, FL  32773

 

Functions, Domain, Range Summary

Domain--Set of all (permissible) x values

Range--Set of all (resulting) y values

To find the Domain, solve for y in terms of x.

To find the Range, solve for x in terms of y.

  

Domain

Page 269. # 53. 

Solution:    Since this equation is in the form , the graphing calculator will probably be a good way to find domain and range.  Just enter this in the calculator.  In the standard window, and the graph should look like this.  In the two sketches below, it is clear that there are points at (-1,0) and (6,0) that will be critical to finding the domain and range.   Notice that the graph actually touches the x axis at these two points, and from these points the graph extends upward from y=0.  The graph extends to the left from x=-1, and to the right from x=6.

            

From this graph, it should be clear that the Domain is .  Likewise, the range is all values that are on or above the x-axis, or , or in interval notation

 

Page 270: #70.  

Solution:  In order to find the domain, you must solve for y in terms of x, in order to see if there are any restrictions on x.  To do this, notice that there is only one y term already on the left side.  You should start by adding  to both sides to get all the non-y terms on the right side .

                           

In order to solve for y, you must divide both sides by ,

                               

                                  

To find the domain, you are looking for restrictions on the denominator.  Denominators can NOT be zero.  In this case, the denominator is  must never be zero, so

Therefore, the domain is all values of  x, such that .

 

Page 271: #75.  

Solution:  In order to find the domain, you must solve for y in terms of x, in order to see if there are any restrictions on x.  To do this, notice that all the y terms are already on the left side, and you can start by factoring out the y.

                           

In order to solve for y, you must divide both sides by ,

                               

To find the domain, you are looking for restrictions on the denominator.  Denominators can NOT be zero.  But in this case, the denominator is , which can never be zero anyway, so there are no restrictions on x. 

Therefore, the domain is all real  x, or

 

 

Range

 

Page 274. # 2. 

Solution:    If you recognize that this is a straight line, then you may already know that the domain and range are both all real values.  However, the general practice in finding the range is to solve for  in terms of  in order to determine what restrictions there might be for   To find the range, solve for .

                                            Subtract  from each side of the equation.

                                         Divide both sides by

                                    or    

                                       or    

Either way you look at it, there are no denominators and no radicals, and therefore there are no restrictions on .  Range is therefore all real , or in interval notation .

 

Page 275. # 18. 

Solution:    If you recognize that this is a parabola that opens upward, then you may already know that the domain is all real values, and the range is .  However, as always, the general practice in finding the range is to solve for  in terms of  in order to determine what restrictions there might be for   To find the range, solve for .

                                              Take the square root of both sides.

                                          

The restriction here is that, because of the square root, the radicand  must be greater than or equal to zero.  That is, the range is , or in interval notation .

 

Page 276. # 25. 

Solution:    You may recognize that this is a parabola that opens downward with vertex at (0, 16)

                                   

Window:                x=[10,10]  y=[-10,20]

                                   

If so, then you already know that the domain is all real values, and the range is all values BELOW the vertex, which would be  or

However, as always, the general practice in finding the range is to solve for  in terms of  in order to determine what restrictions there might be for  To find the range, solve for .

Start by adding to both sides.

                                   

                             

Next add to each side.

                                   

Take the square root of both sides.

                                         

The restriction here is that, because of the square root, the radicand  must be greater than or equal to zero.  That is, the range is

    

Divide by -1:                  

Range:                           

 

Page 276.  # 26.   

Solution:    You may recognize that this is a parabola that opens downward with vertex at (0, −9)

                                   

Window:                x=[10,10]  y=[-20,10]

                                   

If so, then you already know that the domain is all real values, and the range is all values BELOW the vertex, which would be  or

However, as always, the general practice in finding the range is to solve for  in terms of  in order to determine what restrictions there might be for  To find the range, solve for .

Start by adding to both sides.

                                   

                             

 

Next add to each side.

                                   

 

Take the square root of both sides.

                                         

The restriction here is that, because of the square root, the radicand  must be greater than or equal to zero.  That is, the range is

    

Divide by -1:                  

Range:                           

 

 

Page 278. # 35. 

Solution:    Since this equation is in the form , the graphing calculator will probably be a good way to find domain and range.  Just enter this in the calculator, and it should look like this:

                               

From this graph (or from squaring both sides of the equation!), you may recognize this as the upper half of a parabola that extends upward and to the right to infinity.  The range is all values that are on or above the x-axis, or , or in interval notation

The domain (it’s easy, and no extra charge!) just from looking at the graph is

 

Page 278. # 38. 

Solution:    The best bet here is to graph the function, since it is in the form .  Just enter this in the calculator, and it should look like this:

                               

From this graph (or from squaring both sides of the equation!), you may recognize this as the lower half of a parabola that extends downward and to the right to infinity.  The range is all values below the x-axis, or , or in interval notation

The domain (it’s easy, and no extra charge!) just from looking at the graph is

Page 280. # 44. 

Solution:    The best bet here is to graph the function, since it is in the form .  Just enter this in the calculator, and it should look like this:

                               

From this graph (or from squaring both sides of the equation!), you may recognize this as the lower half of a circle that extends 3 units down, 3 units to the right, and 3 units to the left.  The range is all values from -3 up to 0,  or in interval notation

The domain (again, it’s easy, and no extra charge!) just from looking at the graph is .

Page 280. # 46. 

Solution:    It will be best is to graph the function, since it is in the form .  Just enter this in the calculator, and it should look like this:

                               

From this graph you probably won’t  recognize this graph, but it doesn’t matter, does it?  It should be clear that the values of y extend from negative infinity up to zero.    The range is therefore in interval notation 

The domain (again, it’s easy, and no extra charge!) just from looking at the graph is .

.

 

Functions, Domain, and Range

Page 283. 

#3.                  

To find the domain, you must solve for  in terms of .  This means you must first get all the terms on one side, and the non-y terms on the other side.  To do this, subtract from each side.

                        

Factor out the :

                         

Divide both sides by 

                      

Because the denominator must never equal zero, this means that

Domain is all.

Also, this IS a function, since it can be expressed in the form   = _____.

To find the range, you must solve for