Question Video: Determining the Domain of a Piecewise-Defined Function given Its Graph Mathematics

Determine the domain and the range of the function 𝑓(π‘₯) = 6, π‘₯ < 0 and 𝑓(π‘₯) = βˆ’4, π‘₯ > 0.

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Video Transcript

Determine the domain and the range of the function 𝑓 of π‘₯ is equal to six when π‘₯ is less than zero and 𝑓 of π‘₯ is equal to negative four when π‘₯ is greater than zero.

In this question, we’re given a piecewise-defined function 𝑓 of π‘₯ and the graph 𝑦 is equal to 𝑓 of π‘₯. We need to use this to determine the domain of 𝑓 of π‘₯ and the range of 𝑓 of π‘₯. So, let’s start by recalling what we mean by the domain and range of a function.

First, the domain of a function is the set of all input values for that function. Second, the range of a function is the set of all output values for our function given its domain. Let’s start by finding the domain of our function 𝑓 of π‘₯. And there’s two different ways of doing this. First, we can just look at the piecewise-defined function 𝑓 of π‘₯. We want to find all of the possible input values of our function. And we can see from the piecewise definition of our function, our function outputs six whenever π‘₯ is less than zero and our function outputs negative four when π‘₯ is greater than zero. These two inequalities are called the subdomains of our piecewise function. They tell us the input values of π‘₯ for which our function corresponds to the given subfunction.

We can see we’re allowed to input any value of π‘₯ less than zero or any value of π‘₯ greater than zero. This is any value of π‘₯ which is not equal to zero. So, we can input any value of π‘₯ which is not equal to zero into our function. And remember, we write the domain as a set. So, the domain of this function is the set of all real numbers minus the set including zero. However, this is not the only way we can find the input values for our function since we’re given a graph of the function.

Remember, in a graph, the π‘₯-coordinate of any point on our curve tells us the input value of π‘₯ and the 𝑦-coordinate tells us the corresponding output of the function. For example, when π‘₯ is equal to five, we can see the point with coordinates five, negative four lies on our graph. Therefore, we can input the value of five into our function. And 𝑓 evaluated at five is negative four. So, five is in the domain of our function.

Another way of saying this is there’s a point of intersection between the vertical line π‘₯ is equal to five and the given graph. This is sometimes called the vertical line test. We consider vertical lines and look for points of intersection. If there’s no points of intersection, then that value does not lie in the domain of our function. We can consider sliding a vertical line across our diagram. For example, if π‘₯ is equal to 1.5, we can see there’s a point of intersection with our graph. So, π‘₯ is equal to 1.5 is in the domain of our function. We might be worried about values of π‘₯ greater than nine since it appears that this does not intersect our function. However, we can notice the end of our graph has an arrow, and this notation means that our graph continues infinitely in this direction.

The same will be true of the other arrow on our diagram. So, any vertical line on the positive part of our π‘₯-axis will intersect the graph. In fact, the same is true on the negative part of our π‘₯-axis. Any vertical line will intersect the graph. However, we’ve not considered what will happen when our value of π‘₯ is equal to zero. We can see in our diagram there are two points which appear to be on our graph when π‘₯ is equal to zero. However, both of these are hollow circles, and these mean that our function is not defined at this point. So, the line π‘₯ is equal to zero does not intersect our graph, so zero is not in the domain of our function. Therefore, we’ve shown graphically the domain of our function is the set of all real values excluding zero.

Let’s clear some space and then determine the range of our function. Remember, that’s the set of all possible output values of our function given the domain. And once again, there’s two different ways of doing this. We can do this directly from the piecewise definition of 𝑓 of π‘₯. We can see when our input value of π‘₯ is less than zero, our output is a constant value of six. And when π‘₯ is greater than zero, our output value is a constant value of negative four. So, in fact, there are only two possible output values of our function. And the range is the set of these possible output values. The range of our function is the set containing negative four and six.

We could stop here; however, it is also important to be able to determine the range of a function from its graph. This time, remember, the output values of our function are represented by their 𝑦-coordinates on the graph. So, we can determine the range of a function by determining all possible 𝑦-coordinates of points on the curve. However, we can see in our diagram, there are only two possible 𝑦-coordinates of any point on the curve. Either it has a 𝑦-coordinate of six or it has a 𝑦-coordinate of negative four, confirming that the range is the set containing negative four and six.

Therefore, we were able to show for the function 𝑓 of π‘₯ is equal to six when π‘₯ is less than zero and 𝑓 of π‘₯ is equal to negative four when π‘₯ is greater than zero, the domain of this function is the set of real numbers excluding zero and the range of this function is the set containing negative four and six.

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