Functions

In Exercises 7 and...

Question

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b.

Accepted Answer

Welcome back, everyone. In this problem, we want to figure out which of the following graphs represents a function of it. Now how do we know if a graph represents a function? Well, we can use the vertical line test, OK? As we can use the vertical line test. Let me just make a note of that here. And what the vertical line test basically tells us is that a graph represents a function if any vertical line drawn through it intersects at most once. If a vertical line crosses the graph more than once, then it is not a function. And that is because each X value has an output Y value. So if an X, if an input X value has two output values or two y values, then it won't be a function. So based on that definition, let's look through our graphs and see which of these show where a vertical line never intersects the graph twice. If we can find it at least once, that means the graph is not a function. Let's start with our first graph or a circle. Now, this is a graph that has a circle center at the origin with a radius of approximately 10. Now if we were to draw a vertical line anywhere in our graph, let's say we drew one here or we drew one here, notice that in either case it intersects our graph at least twice. But since it intersects it twice at least one time, then we know that this graph is not a function because it fails the vertical line test. Let's move on to our second graph. Will it pass the vertical line test? Well, no, because again we can find at least one point or at least one place on the graph where a vertical line intersects the graph twice. So neither is this second graph a function. Now if we go down to our 3rd graph, OK, this graph consists of a single continuous loop, single continuous curve, sorry, that loops back on itself. So here, notice that if we were to draw a vertical line again. It crosses our graph. Well, in this case 3 times, but because it crosses our graph at least twice, that's how we know that this one is not a function, this 3rd graph. Now how about our 4th graph. This graph consists of two linear segments forming a V shape. Now if you look on the graph, really there's nowhere we can find the vertical line intersecting it twice. For example, if we draw a vertical line here, it doesn't intersect it twice, neither does it here. OK, uh, neither does it here, even at the part where the V is. So what does this tell us? Well, no, we know that this is a function because it passes the vertical line test. Therefore, the fourth option represents a function of X. Thanks a lot for watching everyone. I hope this video helped.

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b. <IMAGE>

Key Concepts

Definition of a Function

Vertical Line Test

Domain and Range

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