If `ZZ` be the set of integers, prove that the function `f: ZZ rarrZZ` defined by `f(x)=|x|`, for all `x in Z` is a many -one function.

Prove that the function f: RR rarr RR defined by, f(x)=sin x , for all x in RR is neither one -one nor onto.

Show f:R rarr R defined by f(x)= x^2 + x for all x in R is many one.

Show that the function f:ZZ rarrZZ defined by f(x)=2x^(2)-3 for all x in ZZ , is not one-one , here ZZ is the set of integers.

Let the function f:R to R be defined by f(x)=x+sinx for all x in R. Then f is -

Prove that, the function f: RR rarr RR defined by f(x)=x^(3)+3x is bijective .

The mapping f:ZZ rarr ZZ defined by , f(x)=3x-2 , for all x in ZZ , then f will be ___

Prove that the operation ** on ZZ defined by a**b=a|b| for all a,binZZ is a binary operation

Let A be the set of quadrilaterals in a plane and RR^(+) be the set of positive real numbers. Prove that, the function f: A rarr RR ^(+) defined by f(x) = area of quadrilateral x, is * many-one and onto.

Show that the function f defined by f(x)= |1-x+|x|| , where x is any real number, is a continuous function.

Let ZZ be that set of integers and f:ZZ rarr ZZ be defined by f(x)=2x, for all x in ZZ and g: ZZ rarr ZZ be defined by, (for all x in ZZ) g(x)={((x)/(2) " when x is even" ),(0" when x is odd" ):} Show that, (g o f) =I_(ZZ) , but (f o g) ne I_(ZZ) .