the class so that 51 can not be image of any element of X under f Hence fis not onto Example 8 Show that the function f N N given by f x 2x is one one but not onto Solution The function f is one one for f x f x 2x 2x x x Further fis not onto as for 1 N there does not exist any x in N such that f x 2x 1 X Rationalised 2023 24 Example 9 Prove that the functionf R R given by f x 2x is one one and onto Solution f is one one as f x f x 2x 2x x x Also given any real number y in R there exists in R such that f 2 2 1 y Hence f is onto 12 02 f x Y Example 10 Show that the function f N N for every x 2 is onto but not one one O is both one one and onto RELATIONS AND FUNCTIONS x 1 if x is odd x 1 if x is even CERT y f x 2x Example 11 Show that the function f R R defined as f x x is neither one one nor onto Solution Since f 1 1 f 1 f is not one one Also the element 2 in the co domain Ris not image of any element x in the domain R Why Therefore f is not onto X Example 12 Show that f N N given by X Solution fis not one one as f 1 f 2 1 But fis onto as given any y N y 1 we can choosex as y 1 such that f y 1 y 1 1 y Also for 1 N we have f 1 1 f 1 2 1 and f x x 1 1 1 x 1 Published O 9 f x x f 1 1 x 1 Y The image of 1 and 1 under f is 1 Fig 1 4