A **function** is a relation from a set of inputs and a set of permissible outputs where each input is related to exactly one output.

A relation from set X to set Y is called function if each element of X is related to exactly one element in Y.

For example, consider the following sets X and Y.

X = { a, b, c }

Y= { 1, 2, 3, 4 }

Relation from X to Y : { (a,1), (b,2), (b,3), (c,4) }

This relation is not function from X to Y because the element b in X is related to two different elements 2 and 3 in set Y.

Relation from X to Y that is a function is : { (a,1), (b,2), (c,4) }

Or { (a,1), (b,3), (c,4) }

This is function since each element from X is related to only one element in Y.

But it is okay for two different elements in X to be related to same element in Y. This is still a function but it’s just not a one-to-one function.

**Relation: **A relation is a set of ordered pairs.

The **domain** is the set of all x values in the relation.

Domain= { -1, 0, 2, 4, 7 }

**Range **is the set of all y values in the relation.

A function f from set X to set Y is a rule of correspondence that assigns to each element in x in that set X exactly one element y in the set Y.

Let’s look at another relation to check if it’s is a function.

The second condition says each x can have only one y, but it **CAN **be the same y as another x gets assigned to.

This is function as it meets our condition.

A good example can be relate to is students in our science class this semester are set X. The grade they received out in the class is denoted by set B. Each student must be assigned a grade and **can only be assigned ONE grade, but** more than one student can get the same grade.

Check this relation out to determine if it is a function.

It is not , because 3 didn’t get assigned to anything.

Comparing to our example, a student in maths must receive grade.

This is not a function , It doesn’t assign each x with a y.

**Function Notation **

We commonly call function by letters. Because function starts with f, it is commonly used letter to refer to functions.

The left hand side of equation is the function notation. It let us know two things. One is function f and the variable in the function is x.

So we have function called f that has the variable x in it. Using function notation we could then ask the following :

Find f(-2). This mean to find the function f and instead of having as x in it, put a -2 in it.

Another example of function is:

One more thing about function is “ Domain”. Which mean values of set X.

For functions we will be dealing with, there are two rules:

- We can’t divide by zero (denominator (bottom) of a fraction can’t be zero.)
- We can take the square root (or even root ) of a negative number.

When we are asked to find the domain of a function, we can use any value of x as long as the value won’t create an “illegal” situation.

**To find domain of a function**

And for negative’s

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