Give the domain and range of the given relation. Is the relation a function?

Domain:

"The set D of all numbers for which f(x) is defined is called the domain (or domain of definition) of the function f" --(edwards, 2)

Note: Your relation is set up as follows:

(x, f(x)), (x, f(x)), (x, f(x)), (x, f(x))

(1, 3) (2, 5) (3, 5) (4, 3)

The domain D is the set of numbers for which f(x) is defined. There is an f(x) defined for this set of numbers {1, 2 , 3, 4 }

Therefore, domain is the set {1, 2 , 3, 4 }

Range:

"The set of all values y = f(x) is called the range of f..."

Note: Your relation is set up as follows:

(x, f(x)), (x, f(x)), (x, f(x)), (x, f(x))

(1, 3) (2, 5) (3, 5) (4, 3)

The range is the set of all values f(x). f(x) in your relation takes on the values {3, 5, 5, 3}

Function:

"A real-valued function f defined on a set D of real numbers is a rule that assigns to each number x in D exactly one real number, denoted by f(x)."

Note: Your relation is defined on the set D = {1, 2 , 3, 4 } as found above. The numbers in this set are real values, this checks out so far with the definition of function...

"A rule that assigns to each number x in D exactly one real number denoted by f(x)"

Again, D = {1, 2 , 3, 4 }. Each number (x) in this set has been assigned "exactly one" f(x). In other words, no x in D = {1, 2 , 3, 4 } has been assigned no more or less values than exactly one f(x).

1 has been assigned 3 (x has been assigned one f(x))

2 has been assigned 5 (x has been assigned one f(x))

3 has been assigned 5 (x has been assigned one f(x))

4 has been assigned 3 (x has been assigned one f(x))

Thus, I find it safe to conclude that your relation is a function according to the definition of a function.

Now, let's say that instead your relation were (1, 3), (2,5), (3,5) (4,3), (1, 10) (your original relation plus (1,10)).

Would this relation be a function?

D = {1, 2, 3, 4} (all of these are real numbers? yes)

the range is the set {3, 5, 5, 3, 10} (also real numbers)

1 has been assigned 3

2 has been assigned 5

3 has been assigned 5

4 has been assigned 3

1 has been assigned 10

So, has each number x in D been assigned exactly one number f(x)?

No: 1 has been assigned 3 and 10 ( more than one f(x)). This relation is not a function.

Domain = {1,2,3,4}

Range {3,5}

Yes - no shared x values makes it a function