Advertisement Remove all ads

Given Examples of Two Functions F: N → N and G: N → N Such that Gof is onto but F is Not Onto. - Mathematics

Given examples of two functions fN → N and gN → N such that gof is onto but is not onto.

(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Advertisement Remove all ads

Solution

Define fN → N by,

f(x) = x + 1

And, gN → N by,

`g(x) = {(x -1, if x>1), (1, if x = 1):}`

We first show that g is not onto.

For this, consider element 1 in co-domain N. It is clear that this element is not an image of any of the elements in domain N.

∴ f is not onto.

Now, gofN → N is defined by,

`gof(x) = g(f(x)) =g(x + 1) = (x +1) -  1`      [x  in N => (x + 1) > 1]

= x

Then, it is clear that for y ∈ N, there exists ∈ N such that gof(x) = y.

Hence, gof is onto.

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 7 | Page 29
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×