In this problem, we're given three relations in the form of tables, as we see here below,
and we're asked to determine which tables are one-to-one functions.
To test to see if a relation is a one-to-one function, we first have
to test to see if it's a function, and then test to see if it's a one-to-one function.
So, let's review both definitions. A function is a relation in which every
input has exactly one output. Remember, the inputs are the x-values or the domain,
and the outputs are the y-values of the range.
An easier way to think of this is:
a relation is a function if every x has exactly one corresponding y-value.
So, once we know a relation is a function, we can then test to see if it's a one-to-one function.
A one-to-one function is a function in which each input has one unique output. Again, an easier way
to think of this is: a function is one-to-one if every y has exactly one corresponding x.
So, looking at our tables, we first want to determine if they're functions and then if
they're one-to-one functions. If they're not functions, they can't be one-to-one functions.
After you do this for a while, you can probably answer these questions by leaving
it in the table format, but when first learning how to do this, I think mapping
the table makes it a lot easier. So, let's look at this first table on the next slide.
To map this relation, we're going to place the set of x-values or inputs here
and the set of y-values or outputs here, and then map the relation. So, looking at the x-values,
we have three unique x-values: we have negative four, negative two, and positive one.
Even though one occurs twice, we only list it once. These are the unique x-values.
And then, for the y-values, we have three, five, and nine. Again, notice how three occurs twice.
And now to map the relation: when x is negative four, y is positive three. So, we'll draw a
segment from negative four to positive three. When x is negative two, y is positive five, so we draw
a segment from negative two to positive five. When x is one, y is nine, so we draw a segment here,
and then when x is one, y is also positive three, so we draw another segment from one to three.
Now, the function test is to make sure every x has exactly one corresponding y-value. In this form,
we can easily see that the x-value of positive one corresponds to two different y-values:
it corresponds to positive three and positive nine. Therefore, this relation is not a function,
and if it's not a function, it can't be a one-to-one function either.
So, we'll go ahead and say it's not a function and move to the next relation.
Next, we'll go ahead and map our second relation. Looking at the x-values, notice how there's no
repetition, so we have negative four, negative two, one, and three. There is repetition in the
y-values, though. The set of y-values will be negative five, positive two, and zero.
Now, we'll map our relation: negative four maps to negative five,
negative two maps to positive two, positive one maps to zero, and positive three maps to two.
We'll first test to see if it's a function, meaning every x has exactly one corresponding y,
and that is true. Notice every x-value
only has one segment moving toward the right, so this is a function.
Now, we'll see if it's a one-to-one function, meaning every y has exactly one corresponding
x. So, looking at the y-values now, notice how the y-value of positive two actually has two
corresponding x-values. Therefore, while this is a function, it's not one-to-one.
This y-value corresponds to the x-value of negative two and the x-value of positive three.
So, it's a function but not one-to-one.
Now, let's test our third relation. Notice there's no repetition in x-values or y-values,
so the mapping is very straightforward. Negative two
maps to three, negative one maps to four, zero maps to five, and one maps to eight.
So, we'll first see if it's a function, meaning every x has exactly one corresponding y, which it
does. Notice every x-value only has one segment moving toward the right, so it's a function.
And to test to see if it's a one-to-one function, we want to see if every y
has exactly one corresponding x. Looking at the y-values,
notice each y-value is only paired with one x because there's only one segment leaving each
y-value moving toward the left. So, this is a function, and it's also a one-to-one function.
So, you may have picked up on the shortcut here:
if there's no repetition in the y-values or the x-values, then as a table,
the relation would be a one-to-one function. But hopefully, by mapping these, you have a better
understanding of what it means to be a one-to-one function. I hope you found this helpful.