Video Transcript
Let 𝑓, a function from the set of
integers to the set of rational numbers, be given by 𝑓 of 𝑛 equals 𝑛 over
one. What is true about 𝑓? Option (A) 𝑓 is not defined. Option (B) 𝑓 is one to one. Option (C) 𝑓 is onto. Or is it option (D) 𝑓 is a
bijection?
In this question, we are given a
function 𝑓 and we need to determine which of four given statements is true about
this function. Let’s start by looking at the given
function 𝑓. We can see that 𝑓 is a function
from the set of integers to the set of rational numbers. In particular, the codomain of the
function is the set of rational numbers. That means that all of the outputs
of the function must be rational.
We can also note that if we take an
input value of some integer 𝑛, then 𝑓 of 𝑛 is defined to be equal to 𝑛 over
one. We need to think of this output as
an element of the rational numbers. Of course, dividing an integer by
one will not affect its value. So we can say that 𝑓 of 𝑛 is
equal to 𝑛. We sometimes leave the division by
one in the definition to help remind us that the codomain is the set of all rational
numbers. However, this is personal
preference.
Let’s now check the validity of the
four given options, starting with option (A). We want to check if 𝑓 is
defined. To check if a function is defined,
we want to check that 𝑓 is well defined for every possible input value. This means that we want to check
two things. We need to check that we can input
any element of the domain into the function. And we also need to check that this
will always give us an element of the codomain.
We have already shown that both of
these conditions hold true. We can input any integer value, and
the output is equal to itself. We know that this output is also a
rational number since the set of integers is a subset of the set of rational
numbers. Hence, the function is defined, and
the answer is not option (A).
Let’s now move on to option
(B). We need to check if the function 𝑓
is one to one. To do this, we can start by
recalling that we say a function is one to one if each element of the range of the
function corresponds to exactly one element of the domain. We can use this along with the
definition of the function 𝑓 to check if the function is one to one. We will check if two elements in
the range of the function can correspond to two different elements in the
domain.
Let’s say that we have two elements
in the range of the function that are equal: 𝑓 of 𝑛 and 𝑓 of 𝑚. Then, by using the definition of
𝑓, we know that 𝑓 of 𝑛 is equal to 𝑛 over one and 𝑓 of 𝑚 is 𝑚 over one. Since division by one does not
change the value of these integers, we can note that 𝑛 must be equal to 𝑚. Hence, we have shown that for any
two elements in the range of 𝑓 to be equal, they must come from the same element in
the domain. Thus, 𝑓 is a one-to-one
function.
For due diligence, we should also
check the other two options.
Option (C) states that the function
𝑓 is onto. We can recall that we say that a
function is onto if every element in the codomain of the function is mapped onto by
some element in the domain of the function. This is equivalent to saying that
the range of the function is equal to the codomain of the function. To check if 𝑓 is an onto function,
let’s check if some values in the codomain of 𝑓 are in its range.
Let’s start by checking if one-half
is in the range of 𝑓. For one-half to be in the range of
𝑓, there must be some integer 𝑛 that maps onto one-half. So 𝑓 of 𝑛 must be equal to
one-half. We then note that 𝑓 of 𝑛 is equal
to 𝑛 over one. And for this to be equal to
one-half, we must have that 𝑛 is equal to one-half. However, this is not an integer, so
this value of 𝑛 is not in the domain of 𝑓. Hence, one-half is not in the range
of 𝑓, but it is in the codomain of 𝑓. This means that the function 𝑓
cannot be onto.
For the final option, we recall
that we say that a function is a bijection if it is both a one-to-one function and
an onto function. We showed previously that 𝑓 is a
one-to-one function. However, it is not an onto
function. This means that 𝑓 is not a
bijection.
So the only true statement about 𝑓
is option (B), that it is a one-to-one function.
