Math

Functions 

Definition:

In mathematics, a function is a linear map between two sets that associates each element of the first set to a single element in the second set. Suppose that A and B are two sets. If one can define a map that takes an element of A to an element in B, then we have a function. 

In the above diagram the elements of A are the inputs, the elements of B are the outputs, and the arrows represent the function. 

 

Domain and Range of a Function:

The domain of a function is the set of all possible inputs for the function while the range of the function is the set of all possible outputs for the function. A general function is defined as y=f(x), where x represents all possible inputs for f, and y represents the set of all possible outputs of f. When we want to determine the domain of a function, we have to identify the points or intervals at which the function is defined. 

 

Finding the Domain of a Function:

In order to find the domain of a function, one has to find the restrictions on the function. In the following, we discuss the domain of some major types of functions.

Linear Functions: A general linear function has the form f(x)=ax+b, where a and b are constant numbers. Since there are no restrictions on the possible values for x, the domain of a linear function is all real numbers. We use the following notations to represent the domain:

                                                                            

Quadratic Functions: A quadratic function has the following general form:

                                                                                              

where a, b, and c are constants. A quadratic function is defined for all real numbers, therefore the domain is

                                                                                         

Square Root Functions: A square root function is defined as:                                                                                                                                                                                            

                                                                                   

as you see, the square root function is defined for zero or positive x values. So, the domain of the function is given by:

                                                                                               

Note that, we use a closed bracket, when we include the number at the endpoint of the interval and we use an open bracket when we do not include the endpoint of the interval.

Trigonometric Functions: The domain of the sine and cosine functions are all real numbers, therefore:

                                                                                         

One can identify the domain of other trigonometric functions by using the trigonometric identities to write the functions in terms of sine and cosine functions. We will look at the domain of tan(x) in the example below.

Cubic Functions: A cubic function is defined as:

                                                                                       

this function can have all real numbers as input. Hence, the domain is all real numbers.

Example: Find the domain.

a) 

In order to find the domain of f, one has to check for all restrictions on the function. We have a square root in the denominator which can only  have positive numbers or zero as the input. However, since we can not have zero in the denominator, we have the following restriction on the function:

                                                                                                    

solving the above restriction gives the domain of the function

                                                                                                            

b) 

In order to find the domain of f, we use the following trig identity

                                                                                                     

both sine and cosine are defined for all values of x. Therefore, the only restriction on the function is having nonzero numbers in the denominator. Therefore, we have