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Examples(2)

(6) Let A = { – 1, 0, 1, 2 } and B = { 0, 1, 4, 9, 16, 25 } & f : A $\large \rightarrow$ B

Let f ( x ) = x ². Find the image of A.

Soln. f ( –1 ) = ( –1 )² = 1, f ( 0 ) = (0)² = 0, f ( 1 ) = ( 1 )² = 1 & f ( 2 ) = ( 2 )² = 4.

$\therefore$ f ( A ) = { 0, 1, 4 }.

(7) Let A = { – 1, 0, 1 } and B = { 0, 1, 4, 9 } and let f be a function from A to B defined by f ( x ) = x ². Show that f is many-one.

Soln. We have : f ( – 1 ) = ( – 1 )² = 1, f ( 0 ) = ( 0 )² = 0 & f ( 1 ) = ( 1 )² = 1.

Clearly, two distinct elements, namely – 1 and 1 of A have the same image in B.

So, f is many-one.

(8) Let f : Q $\large \rightarrow$ Q , f ( x ) = x ⁴. Show that f is many-one.

Soln. We have : f ( – 1 ) = ( – 1 ) ⁴ = 1 & f ( 1 ) = 1 = 1.

Thus, two distinct elements in Q have the same image in Q.

So, f is many-one.

(9) Let f : N $\large \rightarrow$ N, f ( x ) = 5x. Show that f is one-one.

Soln. f ( 1 ) = 5, f ( 2 ) = 10, f ( 3 ) = 15, f ( 4 ) = 20, f ( 5 ) = 25 . . .

Hence, f is one-one.

(10) Let A = { 1, 2, 3 } & B = { 4, 8, 12, 16 } and let f : A $\large \rightarrow$ B, defined by

f ( x ) = 4x. Show that f is not onto.

Soln. We have, f ( 1 ) = 4, f ( 2 ) = 8 & f ( 3 ) = 12.

Range of f = R $\neq$ B = Co – domain.

So, f is not onto.

(11) Show that : f : N $\large \rightarrow$ N, f ( x ) = 3x is into.

Soln. We have, f ( 1 ) = 3, f ( 2 ) = 6 & f ( 3 ) = 9 . . .

Hence, Range of f = R = { 3, 6, 9, . . . } $\ne$ N = Co – domain.

Hence, f is not onto. ( i.e. f is into function. )

(12) Let Y be the set of all non-zero real numbers ( i.e. Y = R – { 0 } ).

Show that : f : Y $\large \rightarrow$ Y : f ( y ) = $\frac{1}{y}$ is an onto function.

Soln. Since inverse of any non–zero real number is also a non–zero real number.

i.e. R = R – { 0 } = Y = Co – domain.

Hence, f is onto function.