Give example of two functions f : N N and g : N N such that gof is onto but f is not onto. (Hint : Consider f(x) = x + 1 and g(x) = * 20 l x - 1, & if & x > 1 1, & if & x = 1 .

Consider, f(x) = x + 1 and g(x) = * 20 l x - 1, & if & x > 1 1, & if & x = 1 . f(x) = x + 1 1 + 1 x N f(x) 2 x N. Clearly, range of f N [1 Range of f] f is not onto. Now, (gof) : N N such that (gof) (x) = g(f(x)) = g(x + 1) = (x + 1) - 1 [ x + 1 > 1 for all x N] = x x N Range of (gof) = N [gof is identity function] Hence, gof is onto.

Relations and Functions — Class 12 Maths Solution

ncert misc SA NCERT Misc.,Q.7, Page 29

Question

Give example of two functions f : N $\rightarrow$ N and g : N $\rightarrow$ N such that gof is onto but f is not onto.

(Hint : Consider f(x) $=$ x + 1 and g(x)$=$ $\left\{ {\begin{array}{llllllllllllllllllll}{x - 1,}&{if}&{x > 1}\\{1,}&{if}&{x = 1}\end{array}} \right.$

Step-by-step Solution

Consider, f(x)$=$x + 1 and g(x)$=$ $\left\{ {\begin{array}{llllllllllllllllllll}{x - 1,}&{if}&{x > 1}\\{1,}&{if}&{x = 1}\end{array}} \right.$

$f(x) = x + 1 \ge 1 + 1\forall x \in N$
$\Rightarrow$ $f(x) \ge 2\forall x \in N.$

Clearly, range of $f \ne N$ [1$\notin$Range of f]

$\therefore$ f is not onto.

Now, (gof) : N $\rightarrow$ N such that (gof) (x)$=$g(f(x))$=$g(x + 1)
$=$ (x + 1)$-$1 [ x + $1 > 1$ for all x $\in$N]
$=$ $x\forall x \in N$

$\therefore$Range of (gof) $=$ N [gof is identity function]
Hence, gof is onto.

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