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 does not represent a function of X. Now how can we tell if a graph represents a function? How can we know? Well, we can tell by using the vertical line test, OK? And by the vertical line test, what it really tells us, OK, is that a graph represents a function if any vertical line drawn through it intersects at most once. So if a vertical line crosses the graph more than once, then it is not a function. So let's take a good look at each of our graphs and use the vertical line test to determine if it can cross our graph more than once, or intersect our graph more than once. Let's start with the first one, or U-shaped curve. Now notice that this graph opens upward and it has a single smooth curve. And now if I were to draw a vertical line. At any point along our graph, let's say I drew a vertical line right here, OK? It intersects the graph only once, and that would be the same if I drew the line anywhere else, OK? So this graph passes the vertical line test, which means that this graph is a function. Now if we go to the next one, the one to the right of it, this is a shape that looks like a sideways parabola. This graph has both an upper and lower part that mirror each other. If I were to draw a vertical line here, OK, then no, you can tell that in at least one place it intersects our graph twice. Thus that means the graph fails the vertical line test. As a matter of fact, at most x values or vertical line will intersect the graph in two places. Thus, this means that this graph is not a function and is our answer. Now, we can make sure that we're right by looking at the remaining graphs, the remaining options. If we go down to our third option here, OK, a smooth curve that continues without loops, notice that this graph has a gradual curve and it does not loop back on itself. And if we put a vertical line anywhere, let's say we put a vertical line here, notice that again, it only touches our graph once. Finally, for our 4th option, this is a diagonal straight line, and if we draw a vertical line anywhere on our graph, OK, then again, it will only touch it once. That's how we're sure that option B is the correct answer or a second graph. Thanks 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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