Domain and Range of Relations from Graphs
Step-by-Step Solutions
Explore detailed step-by-step solutions and explanations for questions on how to find the domain and range of a relation from its graph. Learn how to determine the domain, range, and whether a relation is a function, with clear examples for better understanding.
Question 1 - Domain and Range of a Relation From a Graph
Consider the graph through points A, B and C. Determine the domain and range of the relation. The graph is shown below:
a) Domain: Points \( A(-8 , -0.5) \) and \( B(4,0) \) have the smallest and largest x-coordinates, respectively. Hence, the domain is: \[ -8 \le x \le 4 \] Both points are defined (closed circles), so we use \( \le \).
b) Range: Points \( C(-3,-5) \) and \( B(4,0) \) have the smallest and largest y-coordinates, respectively. Hence, the range is: \[ -5 \le y \le 0 \] Both points are defined (closed circles), so we use \( \le \).
c) The relation is a function because no vertical line intersects the graph more than once.
Question 2 - Domain and Range of a Relation From a Graph
Consider the graph of a curved relation connecting points A, B, and C. Determine the domain and range of the relation. The graph is shown below:
a) Domain: Points \( A(-2, 4) \) and \( B(4, 6) \) have the smallest and largest x-coordinates. Hence: \[ -2 \le x \le 4 \] Closed circles at A and B, so use \( \le \).
b) Range: Points \( C(2,-2) \) and \( B(4,6) \) have the smallest and largest y-coordinates. Hence: \[ -2 \le y \le 6 \] Closed circles at these points, so use \( \le \).
c) The relation is a function because no vertical line intersects the graph more than once.
Question 3 - Domain and Range of a Semi-Infinite Graph
Consider the graph extending infinitely to the left, ending at a closed point A. Determine the domain and range of the relation. The graph is shown below:
a) Domain: The largest x-coordinate is at \( A(4,2) \). The graph extends infinitely to the left, so: \[ x \le 4 \] Closed circle at A, so use \( \le \).
b) Range: The smallest y-coordinate occurs at points \( B(2,-2) \) and \( C(-2,-2) \). The graph extends infinitely upward, so: \[ y \ge -2 \] Closed circle at \( y=-2 \), so use \( \ge \).
c) The relation is a function because no vertical line intersects the graph more than once.
Question 4 - Domain and Range of a Closed Curve
Determine the domain and range of the relation defined by the closed curve shown below:
a) Domain: Points \( A(-5,-1) \) and \( B(1,-1) \) give: \[ -5 \le x \le 1 \]
b) Range: Points \( C(-2,-3) \) and \( D(-2,1) \) give: \[ -3 \le y \le 1 \]
c) The relation is NOT a function because at least one vertical line intersects the graph at two points.
Question 5 - Domain and Range of a Semi-Infinite Graph
Consider the graph starting at point A and extending infinitely to the right. Determine the domain and range of the relation. The graph is shown below:
a) Domain: The smallest x-coordinate is at \( A(-3, 1.8) \). The graph extends infinitely to the right, so: \[ x > -3 \] Open circle at A, so use \( > \).
b) Range: The largest y-coordinate is at \( B(-2,2) \). The graph extends downward infinitely, so: \[ y \le 2 \] Closed circle at B, so use \( \le \).
c) The relation is a function because no vertical line intersects the graph more than once.