Mada za sehemu hiiRelationsMada 4
- Relations
- Graph of relation
- Domain and range of a relation
- Inverse
A graph of a relation represented by a linear inequality or equation is a visual representation of all the ordered pairs (x, y) that satisfy the given relation.
When you are given a relation between two sets of numbers, its graph is obtained by plotting the ordered pairs (x, y) that belong to the relation on the Cartesian plane (x-y plane).
Consider the following relation
So, the relation R can be written as a set of ordered pairs:
R = { (1, 3), (2, 3), (3, 4), (4, 5) }
Each pair (a, b) means that element a from the first set is related to element b from the second set.
Therefore, the ordered pairs: (1, 3), (2, 3), (3, 4), (4, 5) can all be represented on a graph if plotted on an x-y plane, where elements of the first set are x-values and elements of the second set are y-values.
Graph of R is shown in the following diagram (x-y plane).
Draw the graph for the relation R = {(x, y): y = 2x + 1}, where both x and y are real numbers.
Solution:
The equation y = 2x + 1 is a linear equation and represents a straight line. A straight line passes through infinitely many points.
To draw it, we choose at least two values of x, substitute them into the equation to find y, and plot those points.
Let's choose a few values of x:
| x | y = 2x + 1 |
|---|---|
| -2 | -3 |
| -1 | -1 |
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
Now plot the points: (-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5) Join the points with a straight line.
Let A = {-2, -1, 0, 1, 2} and B = {0, 1, 2, 3, 4}
Let the relation R be defined by y = x², where x ∈ A and y ∈ B.
Solution:
We substitute each value of x from set A into the equation y = x² and see if the result is in set B.
| x | x² = y |
|---|---|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
Ordered pairs in the relation R:
R = {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)}
Plot these points on the graph and connect them smoothly to form a parabola (a curved graph).
When a relation is given by an equation such as y = f(x):
- The domain is the set of all x-values that satisfy the equation.
- The range is the set of all corresponding y-values.
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