Video Transcript
Determine the range of the function represented by the given graph.
In this question, we’re given a graph of a function and we need to use this graph to determine the range of the function. And we can start by recalling what we mean by the range of a function. It’s the set of all output values of that function, given the domain of the function, which is the set of all input values of our function. Since we’re given the graph of this function, let’s start by recalling how we find the output values of a function from its graph.
To do this, we remember that any point on our curve has an 𝑥-coordinate and a 𝑦-coordinate. The 𝑥-coordinate is the input value of the function and the corresponding 𝑦-coordinate is the output of the function. For example, if we call our function 𝑓 and we know that our function passes through the point with coordinates five, zero, we know that 𝑓 evaluated at five must be equal to zero. In particular, this means that zero is an output of our function, so zero is an element of the range. We can see this on the diagram because there is a point on the curve with 𝑦-coordinate zero.
We can continue in this manner. We can see that there is a point on the curve with 𝑦-coordinate one, and this continues all the way up to our 𝑦-coordinate of seven. When the 𝑦-coordinate of our curve is seven, we can see that our diagram shows we have a horizontal arrow pointing to the right. This means that the function continues indefinitely in a horizontal line. The function is defined for all of these values of 𝑥. However, its output is a constant value of seven.
But we’re not done yet. We still need to check all of the output values below zero. We can see that there is a solid dot at the end of our curve with coordinates four, negative one. And since this is a solid dot, this means our function is defined at this point. In other words, four is in the domain of our function and 𝑓 evaluated at four is equal to negative one. Negative one is in the range of our function. And our curve contains all values between these two points, and we can also see there is no 𝑦-coordinate of our curve below negative one. Therefore, the range of our function goes from negative one all the way up to seven and it includes both of these values. And this is the closed interval from negative one to seven.
Therefore, we were able to show the range of the function represented by the given graph is the closed interval from negative one to seven.