# One-to one function

A function f: A → B is said to be one-to one function if
f ( a
1 ) = f (a 2 )
⇒ ( a
1 ) = ( a 2 ), ( a 1 )( a 2 ) ∈ A

Let A = {( a
1 )( a 2 )( a 3 )( a 4 )} and B = {( b 1 )( b 2 )( b 3 )( b 4 )} Example : – Determine if the function given below is one to one.
1) To each state of India assign its Capital
Solution: This is not one to one function because each state of India has different capital.

2) Function = {(2,4),(3,6),(-1,-7)}
Solution : The above function is one to one because each value of range has different value of domain.

3) f(x) = |x|
Solution : Here to check whether the given function is one to one or not, we will consider some values of x (domain) and from the given function find the value of range(y).
 1 1 -1 1 -2 2 3 3 -3 3
From the above table we can see that an element in the range repeats, then this is not a 1 to 1 function.

## How to determine one-to one function from the graph?

Q.1 State which of the following graph shows one to one function and why? Note : For checking 1-to-1 function on the graph, we will use a horizontal test.
Horizontal test : Draw a horizontal line on the graph, if that line cuts the graph in two points then the given graph is not 1-to-1 graph.
Solution : In the 1st graph if we draw a horizontal line then that line cuts the graph at one point only so the 1st graph is 1-to-1 function graph.

In the 2nd graph if we draw a horizontal line then that line cuts the graph at one point only so the 2nd graph is 1-to-1 function graph.

In the 3rd graph if we draw a horizontal line then that line cuts the graph at two points so the 3rd graph is not 1-to-1 function graph.

Q.2 Show that the given function (x+2)/(x-3) = (y+2)/(y-3) is one-to one function.
Solution : (x+2)/(x-3) = (y+2)/(y-3)
(x+2 )(y-3) = (y+2)(x-3) ----(cross multiply)
⇒ xy -3x + 2y - 6 = xy -3y + 2x - 6
⇒ -3x + 2y = -3y + 2x -----( xy and -6 get cancelled out)
⇒ -3x -2x = -3y - 2y ----( bring the like terms together)
⇒ -5x = -5y
⇒ x = y ----( divide both side by negative 5)
So the given function is one-to one function.