Find the domain and range of a relation given by its graph. Examples are presented along with detailed Solutions and explanations and also more questions with detailed solutions.
a) Find the domain,
b) the range of the relation given by the graph shown below,
c) state whether the relation is a function or not.
a) Domain:
We begin by identifying the two points on the graph with the smallest and largest \( x \)-coordinates. These are point \( A(-2, -4) \) and point \( B(4, -6) \) (refer to the graph above).
The domain is the set of all \( x \)-values between the smallest and largest \( x \)-coordinates, inclusive. This is written as:
\[ -2 \leq x \leq 4 \]The inequality symbols are "less than or equal to" (\( \leq \)) on both ends because the closed circles at points \( A \) and \( B \) indicate the relation includes those values of \( x \).
b) Range:
Now we find the coordinates of the two points on the graph with the lowest and highest \( y \)-values. These are point \( B(4, -6) \) and point \( C(2, 2) \).
The range is the set of all \( y \)-values between the smallest and largest \( y \)-coordinates, written as:
\[ -6 \leq y \leq 2 \]Again, the use of \( \leq \) is due to the closed circles at points \( B \) and \( C \), meaning the relation is defined at those \( y \)-values.
c) Is the relation a function?
Yes, the relation is a function because no vertical line intersects the graph at more than one point. This satisfies the vertical line test for functions.
a) Find the domain,
b) the range of the relation given by the graph shown below,
c) state whether the relation is a function or not.
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:
\[ -3 \leq x \leq 8 \]The use of the symbol \( \leq \) on both sides indicates that the relation is defined at points A and B (represented as closed circles on the graph).
b) Range:
Points A and B also have the smallest and largest y-coordinates, respectively. The range is given by the inequality:
\[ -5 \leq y \leq 4 \]Again, the use of the symbol \( \leq \) on both sides shows that the relation includes both endpoints.
c)
No vertical line can intersect the graph at more than one point. Therefore, the relation represented by the graph is a function.
a) Find the domain,
b) the range of the relation given by the graph shown below,
c) state whether the relation is a function or not.
a)
Domain: Points A \((-3, -2)\) and B \((1, -2)\) have the smallest and the largest x-coordinates respectively. Hence, the domain is:
The inequality symbol \(\leq\) is used on both sides because the relation is defined at points A and B (closed circles indicate inclusion).
b)
Range: Points C \((-1, -5)\) and D \((-1, 1)\) have the smallest and the largest y-coordinates respectively. Thus, the range is:
Again, the symbol \(\leq\) is used because the relation includes points C and D (both are marked with closed circles).
c)
There exists at least one vertical line that intersects the graph at more than one point (see image below). Therefore, the relation represented by the graph is not a function.
a) Find the domain,
b) the range of the relation given by the graph shown below,
c) state whether the relation is a function or not.
a) Domain:
Point A \((-3, 0)\) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the right 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 value \( x = -3 \), and is written as:
\[ x \geq -3 \]The symbol \( \geq \) is used because the relation is defined at point 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 upper limit to the y-coordinate.
The range is given by all values greater than or equal to the smallest value \( y = -2 \), and is written as:
\[ y \geq -2 \]The symbol \( \geq \) is used because the relation is defined at \( y = -2 \) (closed circles at B and C).
c)
There is no vertical line that intersects the given graph at more than one point (see graph below). Therefore, the relation graphed above is a function.
a) Find the domain,
b) the range of the relation given by the graph shown below,
c) state whether the relation is a function or not.
a)
Domain: Point \( A(-2, -3) \) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the right 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 value \( x = -2 \), and is written as:
\[ x > -2 \]We use the inequality symbol \( > \) (with no equals sign) because the relation is not defined at point \( A \) (open circle at point \( A \)).
b)
Range: Point \( A(-2, -3) \) 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 upper limit to the \( y \)-coordinate.
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 intersects the graph at more than one point. This satisfies the vertical line test.
For each relation below, find the domain and range and state whether the relation is a function.
