# Find Domain and Range of Relations Given by Graphs

Examples and Questions With Detailed Solutions

Find the domain and range of a relation given by its graph. Questions are presented along with detailed Solutions and explanations and also more questions with detailed solutions.

## Examples with Solutions
## Example 1a) Find the domain and b) the range of the relation given by its graph shown below and c) state whether the relation is a function or not.Solution: a) Domain: We first find the 2 points on the graph of the given relation with the smallest and the largest x-coordinate. In this example the 2 points are A(-2,-4) and B(4,-6) (see graph above). The domain is the set of all x values from the smallest x-coordinate (that of A) to the largest x-coordinate (that of B) and is written as:-2 ≤ x ≤ 4 The double inequality above has the inequality symbol ≤ at both sides because the closed circles at points A and B indicate that the relation is defined at these values of x. b) Range: We need to find the coordinates of the 2 points on the graph with the lowest and the largest values of the y coordinate. In this example, these points are B(4,-6) and C(2,2). The range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:-6 ≤ y ≤ 2 The inequality symbol ≤ is used at both sides because the closed circles at points B and C indicates the relation is defined at these values. c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.
## Example 2Find the a) domain and a) range of the relation given by its graph shown below and c) state whether the relation is a function or not.
Solution: a) Domain: In this example points A(-3,-5) and B(8,4) have the smallest and the largest x-coordinates respectively, hence the domain is given by:-3 ≤ x ≤ 8 The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B (closed circles at both points). b) Range: Points A and B have the smallest and the largest values of the y-coordinate respectively. The range is given by the inequality:- 5≤ y ≤ 4 The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B. c) No vertical line can cut the given graph at more than one point and therefore the relation graphed above is a function.
## Example 3Find the domain and range of the relation given by its graph shown below and state whether the relation is a function or not.Solution: a) Domain: Points A(-3,-2) and B(1,-2) have the smallest and the largest x-coordinates respectively, hence the domain:-3 ≤ x ≤ 1 The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B (closed circles at both points). b) Range: Points C(-1,-5) and D(-1,1) have the smallest and the largest y-coordinate respectively. The range is given by the double inequality:- 5≤ y ≤ 1 The relation is defined at points C and D (closed circles), hence the use of the inequality symbol ≤. c) There is at least one vertical line that cuts the given graph at two points (see graph below) and therefore the relation graphed above is NOT a function.
## Example 4Find the domain and range of the relation given by its graph shown below and state whether the relation is a function or not.
Solution: a) Domain: Points A(-3,0) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the left as x increases. Hence there is no limit to the largest x-coordinate of points on the graph. The domain is given by all values greater than or equal to the smallest values x = -3 and is written as:x ≥ -3 The use of the symbol ≥ at because the relation is defined at points A (closed circle at point A). b) Range: Points B and C have equal and smallest y-coordinates equal to -2. The arrow at the top right of the graph indicates that the y coordinate increases as x increases. Hence there is no limit to the y-coordinate and therefore the range is given by all values greater than or equal to the smallest value y = -2 and is written as:y ≥ -2 The use of the inequality symbol ≥ is due to the fact that the relation is defined at y = -2 (closed circle at B and C). c) There is no vertical line that cuts the given graph at more than one point (see graph below) and therefore the relation graphed above is a function.
## Example 5Find the domain and range of the relation given by its graph shown below and state whether the relation is a function or not.Solution: a) Domain: Points A(-2,-3) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the left as x increases. Hence there is no limit to the largest x-coordinate of points on the graph. The domain is given by all values greater than the smallest values x = - 2 and is written as:x > -2 We use of the inequality symbol > (with no equal) because the relation is not defined at points A (open circle at point A). b) Range: Points A(-2,-3) has the smallest y-coordinate equal to - 3. The arrow at the top right of the graph indicates that the y coordinate increases as x increases. Therefore there is no limit to the y-coordinate. Hence the range is given by all values greater than the smallest value y = - 3 and is written as:y > - 3 The inequality symbol > is used because the relation is not defined at y = - 3 (open circle at point A). c) The graph represents a function because there is no vertical line that cuts the given graph at more than one point. ## More Questions
For each relation below, find the domain and range and state whether the relation is a function.
b) c) d) e) ## Solutions to the Above Questionsa)
## Solutiona) Domain: Points A(-8 , - 0.5) and B(4,0) have the smallest and the largest x-coordinate respectively. The domain is the set of all x values between the smallest x-coordinate (that of A) to the largest x-coordinate (that of B) and is written as:- 8 ≤ x ≤ 4 Since the relation is defined at both points (closed circle) the inequality symbol ≤ is used. b) Range: Points C(-3,-5) and B(4,0) have the smallest and largest y-coordinates respectively. Hence, the range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:- 5 ≤ y ≤ 0 The inequality symbol ≤ is used at both sides the relation is defined at these y values (closed circles). c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point. b) ## Solutiona) Domain: Points A(-2 , 4) and B(4,6) have the smallest and the largest x-coordinate respectively. The domain is the set of all x values from the smallest x-coordinate (that of A) to the largest x-coordinate (that of B) and is written as:- 2 ≤ x ≤ 4 Closed circles at both point A and B hence the use of the inequality symbol ≤. b) Range: Points C(2,-2) and B(4,6) have the smallest and largest y-coordinates respectively. Hence, the range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:- 2 ≤ y ≤ 6 The inequality symbol ≤ is used at both sides the relation is defined at these y values (closed circle). c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point. c)
## Solutiona) Domain: Point A(4 , 2) has the largest x-coordinate. As x decreases (moving left), the arrow at the top left indicates that there is no limit to the smallest value of the x-coordinate of any point on the given graph. The domain is the set of all x values smaller than 4 and is written as:x ≤ 4 The closed circle at point A means the relation is defined at x = 4, hence use of the inequality symbol ≤. b) Range: Points B(2,-2) and C(-2,-2) have the smallest (and equal) y-coordinates. The arrow on the top left indicates that as x decreases (moving left), the y coordinate of points on the graph increases without limit. Hence, the range is the set of all y values greater than or equal to -2 and is given by the inequality:y ≥ -2 The inequality symbol ≤ is used because the relation is defined at y = 4 (closed circle). c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point. d) ## Solutiona) Domain: Points A(-5 , -1) and B(1, -1) have the smallest and the largest x-coordinate respectively. The domain is the set of all x values between - 5 and - 1 and is given by:- 5 ≤ x ≤ 1 The closed circle at points A and B means the relation is defined at x = - 5 and x = 1, hence use of the inequality symbol ≤ at both sides. b) Range: Points C(-2,-3) and D(-2,1) have the smallest and the largest y-coordinates respectively. Hence, the range is the set of all y values between -3 and 1 and is given by:-3 ≤ y ≤ 1 The inequality symbol ≤ is used because the relation is defined at both points (closed circle). c) The relation graphed above is NOT a function because at least one vertical line intersects the given graph at two points as shown below. e) ## Solutiona) Domain: Point A(-3 , 1.8) has the smallest x-coordinate. As x increases (moving right), the arrow at the bottom right, indicates that there is no limit to the largest value of the x-coordinate of any point on the given graph. The domain is the set of all x values greater than -3 and is written as:x > -3 The open circle at point A means the relation is not defined at x = -3, hence use of the inequality symbol >. b) Range: Points B(-2,2) have the largest y-coordinates. The arrow on the bottom right indicates that as x increases (moving right), the y coordinate of points on the graph decreases without limit. Hence, the range is the set of all y values smaller than or equal to 2 and is given by the inequality:y ≤ 2 The inequality symbol ≤ is used because the relation is defined at y = 2 (closed circle at B). c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point. |

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