DETERMINING DOMAIN AND RANGE FROM GRAPH

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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