We want to determine which of the following tables represent functions.
A function is a relation in which every input has exactly one output.
So, looking at these tables, the x-values will be the inputs, and the y-values will be the outputs.
If these tables are functions, we could say that every x-value must have exactly
one corresponding y-value. If this is not true, the table would not be a function.
So, we want to analyze these tables and make sure
every x-value has exactly one corresponding y-value.
If we take a look at this first table, notice that the input of two occurs twice
with two different outputs. Here, we have an input of two with an output of three,
and here we have an input of two with an output of negative one.
This does not satisfy the definition of a function because the x-value of two actually
has two different outputs or two corresponding y-values. Therefore, this would not be a function.
To help visualize this, it's often helpful to show the relationship between x and y
using mapping, which means we put the set of inputs, or the domain, here and the set of
outputs, or the range, here, and then draw arrows to show each input with the corresponding output.
So, notice our inputs would be two, negative one, negative four, and five.
We would not list two twice, so we have two, negative one, negative four, and five.
The outputs are the y-values, so we have three, four, zero, negative one, and seven.
When the input is two, the output is three, so we show that relationship by
drawing an arrow from two to three. When the input is negative one, the output is four,
and we'll often say negative one maps to four. We could say negative four maps to zero,
but notice how two also maps to negative one, so we draw another arrow from two to negative one.
And then five maps to seven.
Notice that two has two outputs, one here and one here,
and this is the reason why this is not a function.
Now, let's take a look at the second table. Notice that none of the inputs here repeat themselves.
Therefore, there's no way every x can have more than one corresponding y-value,
so this will be a function, but we'll also map it to illustrate it better.
Again, we'll put the inputs, or the elements or numbers in the domain, here,
and then we'll put the elements or numbers in the range here.
The inputs would be four, negative three, one, two, and five. Notice there is some
repetition in the outputs—there are only three outputs: negative five, negative two, and zero.
Now, let's go ahead and show the mapping. With an input of four, the output is negative five,
so four maps to negative five. Negative three maps to negative two. One also maps to negative two.
Two also maps to negative two. And five maps to zero.
Now, this does look strange because we have three arrows all pointing to negative two in the range,
but notice how every input, or every value here, does have exactly one output, even though at
times they do have the same output. So, because every x has exactly one y, this is a function.
Now, let's take a look at one more example here. The first thing we should notice in
this table is that the input of negative one does occur twice, so negative one does have
two outputs or corresponding y-values. Therefore, again, this table is not a function.
Let's go ahead and map this as well. The inputs are zero, negative one, four, and two. Again, we
do not list negative one twice. Then the outputs will be three, four, zero, negative one, and one.
Zero maps to three, negative one maps to four, four maps to zero, two maps to negative one,
and then negative one maps to one. So, from negative one, we have another arrow going from
negative one to positive one.
Again, because this input of negative one has two outputs, this table does not represent a function.
I hope you found these examples helpful.