Consider functions f and g such that composite gof is defined and is one one. Are f and g both necessarily one-one.

Are f and g both necessarily onto, if gof is onto?

Let f: A to B and g: B to C be two functions. Then; if gof is onto then g is onto; if gof is one one then f is one-one and if gof is onto and g is one one then f is onto and if gof is one one and f is onto then g is one one.

Statement-1 : If f(x) and g(x) are one-one functions then f(g(x)) and g(f(x)) is also a one-one function. and Statement-2 The composite function of two one-one function may or many not be one-one.

If functions f:A to B and g : B to A satisfy gof= I_(A), then show that f is one-one and g is onto.

Which of the following statements are incorrect? I . If f(x) and g(x) are one-one then f(x)+g(x) is also one-one. II . If f(x) and g(x) are one-one then f(x)dotg(x) is also one-one. III . If f(x) is odd then it is necessarily one-one? Ia n dI Ion l y b. I Ia n dI I Ion l y c. I I Ia n dIon l y d. I ,I Ia n dI I I

Let f: A->A be a function such that fof=f . Show that f is onto if and only if f is one-one. Describe f in this case.

If f and g are one-one functions, then a. f+g is one one b. fg is one one c. fog is one one d. none of these

A function f: R to R is defined as f(x)=4x-1, x in R, then prove that f is one - one.

Show that if f : A ->B and g : B ->C are one-one, then gof : A ->C is also one-one.

If fa n dg are one-one functions, then (a) f+g is one one (b) fg is one one (c) fog is one one (d) non e of t h e s e