To determine whether h(fg) is an odd function given that g is an odd function, we first need to recall the definition of an odd function. A function f(x) is considered odd if it satisfies the condition f(-x) = -f(x) for all x in its domain.
Given that g is an odd function, we have:
g(-x) = -g(x) for all x.
Next, we need to analyze the function fg, where f is another function possibly affected by the odd nature of g. If we assume that f is any function, let’s consider the expression:
h(fg):
If we want to show that h(fg) is odd, we need to check if h(fg(-x)) = -h(fg(x)) holds true.
Now, substituting -x into our function:
fg(-x) = f(-x)g(-x)
= f(-x)(-g(x)) = -f(-x)g(x)
If h is a function that preserves the oddness and if f(-x) maintains the oddity, then h(fg(-x)) should yield:
h(fg(-x)) = h(-f(-x)g(x)) = -h(fg(x)),
which confirms that h(fg) is an odd function.
However, if f itself does not enhance the odd nature (if f is not an odd function), we cannot guarantee that h(fg) remains odd without further conditions on h or f.
In conclusion, for h(fg) to be an odd function when g is guaranteed odd, additional conditions on either f or h are necessary. Specifically, if f is also an odd function and h maintains oddness, h(fg) will be odd.