We're asked to classify each function
as one-to-one, onto, both or neither.
We will assume we have functions mapping
from real numbers to real numbers.
A function is a relation in which each input
has exactly one output.
And we are told these four graphs do represent functions.
If we wanted to verify this,
we could use the vertical line test.
The vertical line test indicates
if a vertical line intersects the graph
in more than one point,
then the graph does not represent a function.
If it doesn't, then it does represent a function.
So notice how for our four graphs,
if we pass vertical lines across the graphs,
never will a vertical line intersect a graph
in more than one point,
which verifies all of the graphs do pass
the vertical line test and do represent functions,
which means every input has exactly one output.
And now an easy way to determine whether a function
is one-to-one is to use what's called
a horizontal line test.
If we pass a horizontal line across a function
and it never intersects the graph in more than one point,
then the graph does represent a one-to-one function.
If it does intersect the graph in more than one point,
then it's not one-to-one.
So if we pass horizontal lines across our graphs,
notice how for the first two graphs,
the horizontal lines do intersect the graph
in more than one point.
These two functions are not one-to-one functions,
but horizontal lines will never intersect
the two graphs on the right in more than one point.
And therefore the two graphs on the right
are one-to-one functions.
More specifically,
if we have a function that maps from A to B,
a function is one-to-one or injective
if different inputs have different outputs.
Or we can say every possible Y value in B
has at most one corresponding input X in A.
We can also say outputs or Y values are not repeated.
For the two graphs on the left,
notice how Y values are repeated,
which is why the horizontal line intersects the graph
in more than one point.
We can also say there are Y values
that have more than one corresponding X value,
wherever the horizontal line intersects
the graph in more than one point.
For example, the graph on the left,
these horizontal lines intersect the graph in two points,
which means there are two X values
that give the same Y value.
And notice on the second graph,
there are Y values where there are three inputs of X values
that give the same output or Y value,
which is why these two graphs are not one-to-one.
Similarly, for the two graphs on the right,
for every possible Y value,
there will be exactly one corresponding input or X value,
which is why they are one-to-one functions.
And now let's talk about onto functions.
An easy way to determine whether a function is onto
or subjective is to determine
whether the graph reaches all the Y values in B,
which in our case, B is all real numbers.
So in our case if our graph does not hit all the Y values
on the y-axis, it is not onto.
So if the first graph on the left,
notice how the graph does not reach a Y value
of greater than four,
and therefore the function is not onto.
The second graph reaches all the Y values.
Notice how the graph goes up approaching infinity,
down approaching negative infinity along the y-axis
with no holes or breaks and therefore the function is onto.
Next we have a line.
The line goes up forever and down forever without breaks,
and we'll hit all Y values.
And therefore the function is onto.
Similarly for the cubic function on the far right,
again, the graph goes up forever
and down forever without any holes or breaks,
it will hit all of the possible Y values.
And therefore the function is onto.
More specifically, a function is onto or subjective,
if every output in B is the output of some input in A.
Or we can also say every possible Y value in B
has at least one corresponding input X in A.
And because our functions map
from real numbers to real numbers,
the graph must hit all Y values in order for there
to be at least one corresponding input or X.
So going back to our answers.
For the first graph we can say neither.
For the second graphic we can say the function is onto.
And for the third and fourth graphs we can say both.
Looking back at our answers,
we could answer neither for the first function.
The second function we can say the function is only onto.
And for the third and fourth functions,
we can say the functions are both one-to-one and onto.
I hope you found this helpful.