Let f real numbers set right arrow real numbers set be defi

Let f : real numbers set right arrow real numbers set be defined by f(x) : = 2|x|/root over x^2 + 1. Show that f is not 1 -to- 1. Show that f is not onto.

Solution

f(x) = 2|x|/(sqrt(x^2 +1)

a) for x= -2 : f(-2) = 2*|-2|/sqrt5 = 2*2/sqrt5

x= 2 ; f(2) = 2*2/sqrt5

we can see thet f(x) has more than one value of x taking same value of y

So, f(x) is not 1-1 function

b) A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b

every element of the range corresponds to at least one member of the domain which is not true for f(x)

f(x) = 2|x|/(sqrt(x^2 +1)