# One-to one function

A function f: A → B is said to be one-to one function iff ( 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).

**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.

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