Let f denote an odd function and g an odd function. Decide whether the function h(x)=g(x) f(x) is even or odd.
h(-x) = f(-x)g(-x)
f(-x) = -f(x)
g(-x) = -g(x) ---->
f(-x)g(-x) = f(x)g(x) = h(x)
So:
h(-x) = h(x)
g(x)*f(x) is an even function, since both f and g change signs when x is replaced by -x
To determine whether the function h(x) = g(x) f(x) is even or odd, we need to consider the properties of even and odd functions.
Recall that an even function is symmetric with respect to the y-axis, meaning that f(x) = f(-x) for all x in the function's domain. On the other hand, an odd function is symmetric with respect to the origin, meaning that f(x) = -f(-x) for all x in the function's domain.
Given that g(x) is an odd function, we know that g(x) = -g(-x) for all x in its domain. Similarly, f(x) being an odd function implies that f(x) = -f(-x) for all x in its domain.
Now, let's consider the product h(x) = g(x) f(x). To determine its symmetry, we evaluate h(-x):
h(-x) = g(-x) f(-x)
Since both g(x) and f(x) are odd functions, we can substitute the definitions of odd functions:
h(-x) = -g(x) -f(x)
Next, we simplify by factoring out the negative sign:
h(-x) = -[g(x) f(x)]
Notice that the expression inside the square brackets is equal to -h(x). Therefore, we can write:
h(-x) = -h(x)
This means that h(x) is an odd function since its value at -x is the negation of its value at x.
In conclusion, the function h(x) = g(x) f(x) is odd.