Determine if the following graph shows a function.
Answer
Nyet (no). We can draw a vertical line and have it hit the graph in 3 places. That's overkill, since we should have had our answer the moment we hit the graph a second time, but no harm, no foul.
Example 2
Determine if the following graph shows a function.
Answer
Yes, this is a function. Any vertical line we draw will hit no more than one of the points in the relation. Even those two who are trying to hide along one of the axes. We see ya, fellas. Nice try.
Example 3
Determine if the following graph shows a function.
Answer
This is a function. Any vertical line will hit the graph no more than once. And some will fall right between the two segments of the graph. Missed it by that much.
Example 4
Determine if the following graph shows a function.
Answer
No, this is not a function. If we draw a vertical line through x = 2, we'll hit two different y-values. Only one little rule, graph, and you couldn't follow it? Man...you don't deserve to be a function.
Example 5
Determine if the following graph shows a function.
Answer
Nope, this isn't a function. There's a place where we can draw a vertical line and hit two different y-values. Just because part of this graph is a line and part of it appears in the form of a point doesn't get it off the hook. In fact, we're doubly upset with it for trying to be sneaky.
Example 6
Graph the following relation. Determine, using the vertical line test, if the relation is a function: x = 2 for all values of y.
Answer
This graph is a vertical line through x = 2:
Since a vertical line drawn through x = 2 hits every point in this infinite relation, the relation is not a function. In fact, it couldn't be less of a function if it tried. It's almost as if someone told him he should be a function, and he's acting out in utter defiance. Or maybe we're attributing too many human emotions and behaviors to non-thinking entities. It's been known to happen.
Example 7
Graph the following relation. Determine, using the vertical line test, if the relation is a function: y = -2x.
Answer
Any vertical line will hit the graph no more than once. This relation is a function. It's lucky we're not performing the diagonal line test.
Example 8
Graph the following relation. Determine, using the vertical line test, if the relation is a function: the set containing all pairs of the form (x, y) where x is a positive integer and y is an integer greater than x.
Answer
If we draw a vertical line at any positive integer, we'll hit more than one point in the relation. This relation has a serious point infestation problem, and is clearly not a function.
Example 9
Graph the following relation. Determine, using the vertical line test, if the relation is a function: y = -x2.
Answer
If we draw a vertical line at any positive integer, we will hit only one point in the relation. This relation is a function.
Example 10
Graph the following relation. Determine, using the vertical line test, if the relation is a function: y = 3 for every value of x.
Answer
This graph is a horizontal line through y = 3:
Since a vertical line will hit this relation only once, it is indeed a function. In the world of functions, horizontal lines are our friends. We might even ask them to sign our yearbook.
