Determine which of the following functions are onto If a par

Determine which of the following functions are onto. If a particular function is not onto, find an element of the codomain that is not in the range. F: R rightarrow R given by f(x) =1-x^2 g: R rightarrow [1, + infinity) given by g(x) = x^2 +1 h: R rightarrow R given by h(x) = [x], where [x] denotes the greatest integer function k: R rightarrow R^+ given by k(x) = In(cos^2 (x) +4)

Solution

a)

x^2>=0

-x^2<=0

1-x^2<=1

Hence function is not onto as it does not take values larger than 1

b)

x^2>=0

HEnce,

1+x^2>=0

Hence function is onto

c)

Not onto

As the function only maps to integer values and no non integer real numbers are in range of this function

d)

cos^2(x)>=0

ln(cos^2(x)+4)>=ln(4)>1

HEnce, function is not onto

As for no x in R

k(x)=1