Question Video: Determining the Range of a Piecewise-Defined Function from Its Graph Mathematics

Determine the range of the function represented by the graph.

02:44

Video Transcript

Determine the range of the function represented by the following graph.

In this question, we’re given the graph of a function and we need to use this to determine the range of this function. To do this, let’s start by recalling what we mean by the range of a function. It’s the set of all possible output values of the function given its domain. And we need to determine this set by using the graph of this function.

So let’s recall how the input and output values of a function relate to its graph. We can do this by remembering when we sketch the graph of a function, the 𝑥-coordinate of a point on the curve represents the input value and the corresponding 𝑦-coordinate represents the output value of the function for that 𝑥-input. In particular, this means the 𝑦-coordinate of any point on the curve is an output value of the function.

Therefore, since the range of the function is the set of all possible output values of the function, it’s also the set of all 𝑦-coordinates of points which lie on the curve. This means we can determine the range of this function by just finding the set of all 𝑦-coordinates of points which lie on the curve.

We can do this from the graph. Let’s start by finding the lowest possible 𝑦-coordinate of a point on the curve. From the diagram, we might be tempted to think that this is negative three. However, this is a hollow dot. So remember, this means the function is not defined at this point. So the function never outputs negative three.

However, we can see from the graph of this function we can get closer and closer to negative three. Therefore, the function can output any value close to negative three from above but not equal to negative three. And since this is the lowest output value of the function, we can also say that the function does not output any value less than or equal to negative three.

Let’s now consider output values greater than negative three. And to do this, we need to recall when there are arrows on our function, this means it continues infinitely in this manner. So because there are arrows on both sides of this function, we know both of these straight lines continue indefinitely. We can then use this to determine all values greater than or equal to negative three are possible output values of the function. We can see this by noting any possible value greater than or equal to negative three is a 𝑦-coordinate of a point which lies on the curve. For example, one is a 𝑦-coordinate of a point which lies on the curve.

The only point we might be worried about is two because there’s a hollow dot on our curve at two. However, we can also see that there is another point on the curve with 𝑦-coordinate two. So two is a 𝑦-coordinate of a point which lies on the curve, which also means it’s an output value of the function.

Finally, since our lines continue indefinitely in both directions, we can also see that these continue up to ∞. So this process will hold for all values bigger than negative three. Therefore, the output values of this function are all values greater than negative three. We need to write this as a set, and this is the open interval from negative three to ∞.

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