If a graph represents a function, how many points of intersection does it have with a vertical line?
To answer this question, we’re going to need to start by recalling what we mean by a function. A function maps every element of one set onto exactly one element of a second set. And since we’re mapping elements from one set to another, we consider the first set to be our input set and our second set to be the output set. So we can think of this definition in terms of inputs and outputs. A function will take an input and give exactly one output. We want to see how this definition compares to the graph of a function and the number of intersections it will have with a vertical line.
To do this, let’s start by recalling how we graph a function. And one way of doing this is to look at an example. Let’s consider the line 𝑦 is equal to 𝑥 plus one. In the graph, the 𝑥-coordinates of points on the curve represent the input values and the 𝑦-coordinates represent the corresponding outputs. So every point on the graph of this function will have coordinates of the form 𝑥, 𝑓 of 𝑥 or alternatively 𝑥, 𝑥 plus one. Let’s now consider a vertical line on our diagram, for example, 𝑥 is equal to negative two. In this diagram, we can see there’s only one point of intersection. And we know how to find the 𝑦-coordinate of this point of intersection. The 𝑥-coordinate of this point is negative two. So the corresponding 𝑦-coordinate of the point of intersection is 𝑓 evaluated at negative two. And negative two plus one is negative one.
And we can notice something interesting. Every intersection between this vertical line and the function gives us an output of the function. This allows us to prove there’s no second point of intersection between the vertical line and the graph of our function, since every point of intersection will have coordinates of the form negative two, 𝑓 evaluated at negative two. And the definition of a function tells us that there’s only one output. We might be tempted to answer our question by just saying there’s one point of intersection. However, we do need to be very careful. Really, we’ve only shown that there’s at most one point of intersection. In fact, we can prove that there doesn’t need to be a point of intersection.
For example, consider the graph of the function 𝑦 is equal to root 𝑥. We know that there’s no point of intersection between the vertical line 𝑥 is equal to negative one and the curve 𝑦 is equal to root 𝑥. And this is because no negative value of 𝑥 is in the domain of our function 𝑓 of 𝑥. However, this function still takes one input value and gives us exactly one output value, so it’s still a function. So it’s possible for the vertical line and graph of our function to not intersect at all, but they can’t intersect two or more times. Therefore, we’ve shown that the graph of a function and a vertical line will intersect at most once.