ONE TO ONE FUNCTION

Example 1 :

Check whether the following function is one-to-one 

f : N → N defined by f(n) = n + 2

Solution :

To check if the given function is one to one, let us apply the rule

f (x) = f (y) ----> x = y

f(x) = x + 2 and f(y) = y + 2

x + 2 = y + 2

x = y

For every element if set N has images in the set N.

Hence it is one to one function.

Example 2 :

Check whether the following function is one-to-one 

f : R → R defined by f(n) = n²

Solution :

To check if the given function is one to one, let us apply the rule

f(x) = f(y) ----> x = y

f(x) = x² and f(y) = y²

By equating f(x) and f(y), we get

x² = y²

x = y

From this we cannot decide that the function is one to one. Because every two different elements in the domain has same images is co-domain. That is,

If x = 1 then y = 1. If x = -1 also, then y is 1.

Hence the given function is not one to one.

Example 3 :

Check whether the following function is one-to-one 

f : R - {0} → R defined by f(x) = 1/x

Solution :

To check if the given function is one to one, let us apply the rule

f(x) = f(y) ----> x = y

f(x) = 1/x and f(y) = 1/y

By applying the above rule, we get

1/x = 1/y 

x = y

Every element in domain has different images in co-domain. 

Hence it is one to one function.

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