2. Determine whether f is a function from Z to R if

First, we need to know the definition of a function.

Definition: “Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f: A → B.” Discrete mathematics and its applications By Rosen.

Now we can solve the exercise.

a) f(n)=±n

f(2)=2, f(2)=-2

From the definition of function “…. A function f from A to B is an assignment of exactly one element of B to each element of A…”.

Therefore, it is not a function.

b)f(n)= √(n²+1)

It is a function because is defined for each element of the domain. Notice that n²+1>0 for all n∈Z and the square root is defined on R for any positive integer.

c) f(n) = 1/(n² − 4)

From the definition of function “…. A function f from A to B is an assignment of exactly one element of B to each element of A…”.

This function is not defined for n=2.

Therefore, it is not a function because is not defined for each element of the domain.

Related exercises:

• 10- Determine whether each of these functions from {a, b, c, d} to itself is one-to-one.
• 6. Find the domain and range of these functions
• 4. Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function.