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Unit 4: Topic2: Writing Domain and Range of Relations from their Graphs

Competencies

·   Sketch graphs of relations

·   Determine the domain and range of relations from their graphs

Suggested ways of teaching this topic: Guided Practice

Starter Activities

The teacher may start with asking questions of the following type:

“Who can define relations?”

“Who will tell us what domain means?”

“What about range?”

Expected Answers:

A relation is a set of ordered pairs

The domain is the set of 1st coordinates of the ordered pairs.

The range is the set of 2nd coordinates of the ordered pairs.

After the definitions are revised, the teacher may ask students to come up with answers to questions of the following type:

Given the relations: R1= {(3,2), (1,6), (-2,0)} and R2= {(2, 4), (3,-1), (0,-4), (3, 4)}find the domain and range of each relation.

Expected Answers:

For R1 , Domain = {3, 1, -2} and Range = {2, 6, 0}

For R2 , Domain = {2, 3, 0} and Range = {4, -1, -4}

Example 1: What is the domain of the relation {(2, 1), (4, 2), (3, 3), (4, 1)}

  1. {2, 3, 4, 4}
  2. {1, 2, 3, 1}
  3. {2, 3, 4}
  4. {1, 2, 3}
  5. {1, 2, 3, 4}

Example 2: What is the range of the relation {(2, 1), (4, 2), (3, 3), (4, 1)}?

  1. {2, 3, 4, 4}
  2. {1, 2, 3, 1}
  3. {2, 3, 4}
  4. {1, 2, 3}
  5. {1, 2, 3, 4}

After a short summary on the definitions of domain and range of relations, the teacher may continue the discussion, using appropriate examples of type given below, to answer the important question: “How could we determine the domain and range of relations with two or more inequalities?” (The graphs could be graphs of those relations discussed in Topic 4 above.

Major Idea:

To determine domain and range of relations with two or more inequalities, it is better to sketch their graph and see the region covered by the relation both horizontally and vertically.

The teacher may start with those graphs of relations with only one boundary line and continue with relations with two or more inequalities.

Lesson Notes

Example: Determine the domain and range of relations represented by the graphs given below

a)

 

b)

c)

d)

Solution:

a)      Since the border line goes non-stop on both directions, there is no limit to the relation (shaded region) vertically and horizontally. Thus, the domain and range of this relation is the set of all real numbers

b)      Same as in a)

c)      As the boundary line does not limit the shaded region horizontally, the domain of the relation is the set of real numbers. But, as the boundary line limits the relation vertically at y = 2 and the line y = 2 is a solid one, the range of the relation is given by {y: y 2}.

d)     As the boundary line limits the relation horizontally at x = -1 and the line is solid, the domain of the relation is given by {x: x -1} But, as the boundary line does not limit the shaded region horizontally, the range of the relation is the set of real numbers.

CONCLUSION

The teacher might ask students to conclude on the domain and range of the relations given by graphs of type a) and b) above. The students are expected to conclude as follows:

If the relation is represented by a graph with shaded region above or below one boundary line which is neither vertical nor horizontal straight line, the domain and range of the relation will be the set of all real numbers.

Example: Determine the domain and range of the relations represented by the graphs below.

a)

b)

c)

d)

Solution:

a)      Since the boundary lines y = 2 – x and y = x + 2 meet at (0, 2), with y= 2 – x a broken line, and the shaded region is limited horizontally to the right by the line x = 0, the domain of the relation is {x: x < 0}. But, as the region is not limited vertically, the range of the relation is the set of all real numbers.

b)      The first step is to determine the intersection point of the boundary lines. To get that, we can solve both equations simultaneously using substitution method. Thus, substituting the value of y = 3 into y= x +1, we get 3 = x +1, which gives x = 2. So, the intersection point of the boundary lines is at (2, 3).

When we see the shaded region, it is limited horizontally to the right by the line x = 2, with the line y = x + 1 broken. So, the domain of the relation is {x: x < 2}. The shaded region is also limited vertically upwards by the solid line y = 3. So, the range of the relation is {y: y  3}.

c)      Observing the shaded region, we see that it is triangular form, where it is bounded horizontally as well as vertically from both directions. Horizontally, the region is bounded by a broken line x = -1 from the left and by a solid line x = 3 from the right. Thus, the domain of the relation is {x: - 1 < x  3}.  To get the boundary lines which limited the shaded region vertically, we need to find the intersection points of x = 3 with the line y = x + 1 and y = - x -1. This can be done by substitution. If we substitute x = 3 in y = x + 1and y = - x -1, we get y = 4  and y = - 4 respectively. Therefore, the range of the relation is {y:  -4 < y < 4}.

Note:

Students must be reminded about the type of relationship between the strict less or greater symbols and broken the lines as well as why that relationship existed.

d)     This shaded region is also in a triangular shape, bounded both horizontally and vertically. We need the intersection points of the lines x = 2 and y = 3 with the line y = 1– x. So, substitution results in y = -1 and x = – 2 respectively. So, the intersection points are (-2, 3) and (2, -1). Thus, the domain of the relation is {x: -2 < x < 2} and its rangeis {y: -1 < y < 3}.

Concluding Activities

As discussed above, determining the domain and range of relations given their graphs is to see where the relation is limited vertically or horizontally or both. Students must be given relations without graphs, say R = {(x, y): y £ x - 2, y< -x+3 and y > -4}, and shall be asked to determine the domain and range. (Of course, they need to sketch graphs!).Therefore, the teacher shall make sure that students have become capable of sketching graphs of relations and can write the domain and range from the graphs giving them relations of the above type and guiding them to practice and master the skills.

Practice Exercises

1.                  Write the inequality represented by each of the following graphs and determine their domain and range

a)

b)

c)

d)

2.                  Sketch the graphs of the following relations and determine their domain and range.

A.    R = {(x, y): y > x -1 and x > 2}

B.     R = {(x, y): x + y ≤ 2 and y < x}

C.     R = {(x, y): y ≤ x + 1, x + y ≥ -2 and x < 2}

D.    R = {(x, y) = y <x + 2 and x + y < 0}

E.     R = {x, y) = x<y+2 and y +x < 0}