Question Video: Determing the Domain and Range of a Function from Its Graph Mathematics

Determine the domain of the function represented by the following graph. What is the range of the function?

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Video Transcript

Determine the domain of the function represented by the following graph. What is the range of the function?

In this question, weโ€™re given the graph of a function and weโ€™re asked to determine its domain and its range. And we can start by recalling what we mean by the domain of a function. Itโ€™s the set of all input values for that function. And we need to determine this set from the graph of the function. To do this, we can recall when we graph a function, the ๐‘ฅ-coordinate of any point on the curve represents the input value of our function and the corresponding ๐‘ฆ-coordinate tells us the output value of the function for the corresponding value of ๐‘ฅ.

For example, we can see that the graph of our function passes through the point with coordinates two, three. So if we call our function ๐‘“, this tells us the ๐‘“ evaluated at two is equal to three. And in particular, this means we can input the value two into our function. Two is in the domain of our function ๐‘“.

Therefore, another way of thinking about the domain of a function is the set of all ๐‘ฅ-coordinates of points on its graph. For example, we can see thereโ€™s a point on the graph of ๐‘ฅ-coordinate two. So two is in the domain of our function. In fact, we can see this is true for any value of ๐‘ฅ we choose. For example, if we chose ๐‘ฅ is negative three, we can see thereโ€™s a point on the graph of ๐‘ฅ-coordinate negative three. So negative three is also in the domain of our function.

Another way of thinking about this is any vertical line will intersect the graph of our function. Therefore, the domain of our function is any real value of ๐‘ฅ. We write this as the set of real numbers.

Next, we need to determine the range of this function. And we can start by recalling the range of a function is the set of all output values for the function, given its domain, which is the set of input values. To determine the range of this function from its graph, letโ€™s start by clearing some space.

We recall the ๐‘ฅ-coordinate of any point on the graph tells us the input value and the corresponding ๐‘ฆ-coordinate tells us the output value of the function for that value of ๐‘ฅ. For example, we saw since the point with coordinates two, three lies on our graph, ๐‘“ evaluated at two is equal to three. We use this to determine that two was in the domain of our function ๐‘“. However, we could also see that three is in the range of our function ๐‘“. Three is a possible output of our function.

Another way of saying this is there is a point on the curve with ๐‘ฆ-coordinate three. Therefore, another way of thinking of the range of a function is the set of ๐‘ฆ-coordinates of points which lie on our curve. For example, we can see that there are two points on the graph of our function with ๐‘ฆ-coordinate four. Therefore, four is a possible output of our function. Four is in the range of ๐‘“.

And since the graph of our function continues to positive โˆž on both sides, we can see this will be true for any value of ๐‘ฆ greater than or equal to three. However, thereโ€™s no point on the graph of this function with ๐‘ฆ-coordinate less than three. For example, we can see thereโ€™s no point on the graph of ๐‘ฆ-coordinate negative two. So itโ€™s not possible for our function to output a value less than three. Therefore, the output values of our function is all values greater than or equal to three.

And remember, the range is a set. So we need to write this in set notation. We start with three because itโ€™s the lowest value in our set, and we use the closed bracket because we need to include this value. Then we want to include all values up to โˆž. So we end with an โˆž symbol and an open bracket because we donโ€™t include this value. This then gives us our final answer. The domain of the function represented in the graph is the set of all real values โ„, and the range of this function is the set of values greater than or equal to three.

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