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

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

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Video Transcript

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

In this question, we’re given the graph of a function and we’re asked to determine its domain and its range. And we can start by recalling what we mean by the domain of a function. It’s the set of all input values for that function. We need to determine this set from the graph. And to do this, we need to recall what we mean by the graph of a function. When we sketch the graph of a function, we sketch it on a pair of coordinate axes. The 𝑥-coordinate of a point on our graph tells us the input value and the corresponding 𝑦-coordinate tells us the corresponding output.

For example, we can see the point with coordinates seven, three lies on our graph. If we call our function 𝑓 of 𝑥, then this tells us that seven is an input value of our function and the corresponding output is three. In other words, 𝑓 evaluated at seven is equal to three. And we want to determine all of the input values of our function, so this tells us that seven is an input value for our function. Seven is in the domain of 𝑓 of 𝑥. Another way of thinking about this is the 𝑥-coordinates of the points on our curve tell the input values. So, for example, we can see a point with 𝑥-coordinate four lies on our curve. Four is also in the domain of our function. We need to determine all of the input values for our function, so we need to determine all of the possible 𝑥-coordinates of points which lie on this curve.

To do this, we can start by noticing our curve stops at 𝑥 is equal to negative two. We can see negative two lies on our curve. However, there’s no point on the curve with 𝑥-coordinate less than negative two. So all values less than negative two will not lie in the domain of our function. Next, it’s important to realize that the graph of this function is going to continue off infinitely, and this means that we can take any input value of 𝑥 greater than or equal to negative two. Therefore, the domain of this function is the set of all values greater than or equal to negative two. We need to write this in set notation. We want to include the value of negative two. So we start with a square bracket, and we want to go all the way up to ∞. However, this has an open bracket because we don’t include this value. This then gives us the domain of our function.

Now let’s move on to the second part of our question. We want to determine the range of this function. We recall the range of a function is the set of all output values of the function given its domain. To determine this set from the given graph, let’s start by clearing some space. We’ve already seen that the output values of the function are given by the 𝑦-coordinates of any point on the graph. For example, we saw since the point with coordinates seven, three lies on our graph, if we label our function 𝑓, we have 𝑓 evaluated at seven is equal to three; three is an element of the range of our function. Therefore, the range of a function is the set of all the 𝑦-coordinates of points which lie on its graph.

We can then determine the range of a function graphically. For example, we can see that two is in the range of our function. There is a point on the curve of 𝑦-coordinate two, but we need to determine all of the 𝑦-coordinates. In this case, we can see that there is a point with lowest 𝑦-coordinate. The point with coordinates negative two, zero lies on our curve. And there’s no point with lower 𝑦-coordinate which lies on our curve. So zero is in the range of our function and there’s no lower value than zero in its range. To determine the rest of the range of this function, we again need to remember that our curve is going to continue off infinitely.

And in the graph, it appears we’re not given a horizontal asymptote of our function, so we can conclude the function is unbounded. Any positive value of 𝑦 will also be in the range. So the range of this function contains all values greater than or equal to zero. And remember, we need to write this as a set. We want to include the value of zero, so we use a square bracket. And we want to go all the way up to ∞, which we write with an open bracket because ∞ is not included. This thing gives us our final answer. The domain of the function represented in the graph is the set of all values greater than or equal to negative two, and the range of the function is the set of all values greater than or equal to zero.

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