Key Ideas :
• The domain of a function is the set of values of the independent variable for which the function is defined. The range of a function depends on the equation of the function. The graph depends on the domain and range.
• The domain and range of a function can be determined from its graph, from a table of values, or from the function equation. They are usually easier to determine from a graph or a table of values.
For each relation, state the domain and range and whether the relation is a function.
Example 1 :
Solution :
In the graph above, x-coordinates are all the integers from -3 to 3 and the y-coordinates are all the integers from -2 to 4.
Domain = {x ∊ Z | -3 ≤ x ≤ 3} or or {-3, -2, -1, 0, 1, 2, 3}
Range = {y ∊ Z | -2 ≤ x ≤ 4} or {-2, -1, 0, 1, 2, 3, 4}
The graph passes the vertical-line test. So, the graph is a function.
Example 2 :
Solution :
An open circle on the graph shows that the endpoint of the line is not included in the graph. A closed circle means that the endpoint is included. So, x cannot be -5, but it can be 11.
Domain = {x ∊ R | -5 < x ≤ 11}
Range = {2, 6}
The graph passes the vertical-line test. So, the graph is a function.
Example 3 :
Solution :
The graph is a parabola with a maximum value at the vertex, which is the point (1, 3).
Therefore, x can be any real number, but y cannot be greater than 3.
Domain = {x ∊ R}
Range = {y ∊ R | y ≤ 3}
The graph passes the vertical-line test. So, the graph is a function.
Example 4 :
Solution :
The graph is a circle with center (0, 0) and radius of 5.
Domain = {x ∊ R | -5 ≤ x ≤ 5}
Range = {y ∊ R | -5 ≤ y ≤ 5}
The graph fails the vertical-line test. Because, there are many vertical lines that cross the graph in two places. So, this is not a function.
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