How To determine if a function is one to one (1-1)

Functions are the base of mathematics and Science. In simple words if we have two sets X and Y of some items then the function can be described as a procedure or a rule that assigns one item of X to another item in Y. For example the rule that squares numbers takes members of the set X={1,2,3,4} to corresponding elements of Y={1,4,9,16}. Mathematically it can be written as  f : X→Y, where f is the function, X is tdomain and Y is Range.

Now depending on the rule or procedure, functions have many types, one of which is one-to-one that we will be discussing. In simple definition “a function is said to be one-to-one if no two different elements of X yield the same output in Y” , so the example given above can also be classified as one-to-one function.
Let's see two methods to find if the function is one to one or not.


   

Method  1 of 2

To show that a function is one-to-one, we could show that

f(x) = f(y)   =>     x = y

Step 1:  Take the function f(t) you want to check, e.g.

f(t) = t-3/t+2

Step 2:  Make functions  and   from  by changing the variable t to x and y respectively, so that we can perform the test with two different variables:

 f(x)=x-3/x+2                       f(y)=y-3/y+2  

Step 3: Equate both the functions as   f(x)=f(y)  ,

x-3/x+2  =  y-3/y+2 

Step 4: Simplify the above equation. Now we need to solve the equation to its simplest form. To do this we will cross multiply the terms.After that we will multiply the terms by opening the brackets like. If we further simplify it as shown below then in the end we will get,

               

2x−3y          =       −3x+2y

2x+3x         =         2y+3y

                                                                   5x               =              5y

                                                                    x                =              y

Step 7 : Check whether x = y or not ,if in the end we come up with  x=y (as in the example)  then we have shown that the function is one-to-one if not then the function is not one-to-one.

Method 2 of 2

Step1 : Get the function f(x) you want to check. This is a graphical method for determining that if a function is one-to-one. To apply this method get a function f(x) you want to check e.g.

f(x) = x-3/x+2

Step2 : Plot the graph of the function f(x) on the x-y plane. You can either draw it on the paper or any mathematics software. To draw a graph on the paper first calculate some values of f(x) for few numbers like  x=…..,-5,-4,-3,-2,-1,0,1,2,3,4,5……  and plot them on the paper. After you join these pairs you will get an estimated shape of the function like the graph shown below:

Step 3: Draw a horizontal line anywhere on the graph to perform the horizontal line test which is:

The function is one-to-one if every horizontal line drawn anywhere in the x-y plane intersects the graph in at most one point. If it intersects the graph at more than one point then the function is not one to one. For example let’s apply this test on the above graph as shown below:

Since the horizontal line intersects at only one point so it’s a one-to-one function.

Tips:

·         Practically speaking functions are really important in science and engineering. For example a function can represent a physical system and by checking whether a functions one-to-one we can tell whether the system is reliable, stable or even implementable.

·         The second method is graphical and is unreliable because some functions exist that show one to one behavior for smaller values of x but as the values shoot to higher values it starts diverting.