Question

The entire graph of the function f is shown in the figure below. Write the domain and range of f using interval notation?

Answer 1

Domain:   (-5, 3]

Range:  [-4, 5)

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Explanation:

The domain is the set of allowed x values. In terms of a graph, we look at the left most point to see that x = -5 is the smallest x value possible. However, there's an open hole at this endpoint, so -5 itself is actually not part of the domain. So x must be larger than -5. At the same time, x can be as large as x = 3. Look at the very right tip of the graph to find this x value.

So x spans from -5 to 3, excluding -5 but including 3. We would write $-5 < x \le 3$ which converts to the interval notation (-5, 3]. Note the mix of curved parenthesis and square bracket. The curved parenthesis means to exclude the endpoint, while the square bracket means include the endpoint.

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The range is the set of possible y outputs. Find out the lowest point of the graph. That is when y = -4 and this value is included due to the filled in circle at the endpoint. But we do not include the largest y value y = 5 as there's an open hole at this endpoint.

So the range is the set of y values such that $-4 \le y < 5$ which in interval notation would be written as [-4, 5)

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So in short, you're looking for the min and max of x and y to get the domain and range respectively. Be sure to exclude any values where there are open holes as those do not count as part of the graph.