Determine the domain of the following function.
In this question, we’re given the graph of a function. And we need to use this to determine its domain. And we can start by recalling the domain of a function is the set of all input values for that function. So we need to use the graph of the function to determine the set of all input values for the function. To do this, we recall that any point on our curve will have an 𝑥-coordinate and a 𝑦-coordinate. The 𝑥-coordinate tells us the input value of 𝑥 for our function, and the corresponding 𝑦-coordinate tells us the output of the function for that 𝑥-coordinate.
For example, we can see our curve passes through the point with coordinates three, negative four. This tells us that 𝑓 evaluated at three is equal to negative four. And in particular, this tells us that three is in the domain of our function; it’s a possible input value for our function. We want to determine the domain of this function, so we need to determine all of the possible 𝑥-coordinates which lie on our curve.
We can do this in steps. Let’s start by looking at values of 𝑥 bigger than three. We can see as we move to values of 𝑥 greater than three, there are 𝑥-coordinates all the way up to 𝑥 is equal to six. However, when we get to input values of 𝑥 equal to seven, we can see our curve has an arrow. This type of notation means that our curve continues infinitely in this direction. And in particular, since this is a horizontal arrow at 𝑦 is equal to negative four, our curve will just continue horizontally at negative four.
So we can see that there are input values of 𝑥 for all values of 𝑥 greater than or equal to three. We can then do the same for input values of 𝑥 less than three. We can keep taking input values of 𝑥 closer and closer to zero. However, when we get to zero, we notice something interesting. At the point zero, negative four, our curve has a hollow dot. This means our function is not defined at this point. Instead, we need to notice that we have a solid dot at the point with coordinates zero, four. Since this is a solid dot, this means our function is defined at this point. 𝑓 evaluated at zero is equal to four, so zero is in the domain of our function.
We can then continue in the same way for values of 𝑥 less than zero. And when we get to 𝑥 is equal to negative seven, we have the same thing which happens before. The graph has an arrow, so it continues infinitely in this direction. So we can take any input value of 𝑥 less than or equal to zero.
This is enough to determine the domain of our function. On the left, we can see we can take any input value of 𝑥 less than or equal to zero. And on the right, we can see we can take any input value of 𝑥 greater than zero. Therefore, the domain of our function is the set of all values less than or equal to zero or greater than zero. But this is just any possible value. So we can say the domain of the function given in the graph is the set of real numbers ℝ.