Prove that
For g and h being odd functions that g(h(x)) is also an odd function
Let us first define what an odd function is:
A function f(x) is said to be odd if f(-x) = -f(x) for all x in the domain of f.
Now, since g and h are odd functions, we have:
g(-x) = -g(x) for all x in the domain of g
h(-x) = -h(x) for all x in the domain of h
Now, let's consider the function g(h(x)). We want to prove that g(h(-x)) = -g(h(x))
By the definition of an odd function and the properties of g and h as odd functions, we have:
g(h(-x)) = g(-h(x)) (since h is an odd function)
= -g(h(x)) (since g is an odd function)
Thus, we have shown that g(h(x)) is an odd function, as g(h(-x)) = -g(h(x)) for all x in the domain of g(h(x)).