Which of the following functions is odd? and why?

Question

g(x) = 2sin xcos xtan^2x
h(x) = cos x(sin x + 2)
f(x) = 3sin xtan x
s(x) = cos^2x(1 – sin x)

Answer 1

odd functions:
f-x) = -f(x)
g(-x) =2sin(-x)cos(-x)tan^2(-x)

now sin(-x) = -sinx
cos(-x)=cosx
tan(-x)=tanx
so g(-x) = -2sinxcosxtan^2x=-g(x)
odd function

similarly do otherr fns

also graphically if you plot these fns they are antisymmetric wrt zero on coordinate plane

if f(-x) =f(x) then it is evn fn.
while if f(-x) is neither + nor - f(x) but a different function then it is neither odd nor even likes(x)

Answer 2

An odd function is a function f(x) such that f(-x) = -f(x). You need to test each function with -x to see if the result becomes negative. In the case of a product, this holds true if an odd number of the factors becomes negative, but it fails if an even number of them become negative (because the product of negatives is positive).

Answer 3

subtitute for x= -x and if u get the function as -ve then it is odd and if u get the original function after sub. then is is even
1.g(x)=2 sinxcosxtan^2x is odd cause sin(-x) is -sin(x) and cos(-x) =cos(x) and tan^2(-x)=tan^2(x) cause square is present so if u muntiply u will get -ve that means it is odd

2.h(x) is odd
3.f(x) not odd
4. s(x) is odd

For all the questions the reason is same.