give an example of two increasing functions whose product is not increasing
f(x)=i x, g(x)=i(x^2) where i is the sqrt(-1) for x>0
without delving into complex numbers, maybe something like
f(x) = e^2x
g(x) = -e^-x
(fg)(x) = -e^x, which is decreasing
Sure, let's take the functions "Hangover(x)" and "Procrastination(x)". Both of them can be described as increasing functions—Hangover(x) represents the severity of a hangover, and Procrastination(x) represents the amount of time spent procrastinating.
Now, if you take the product of these functions, Hangover(x) * Procrastination(x), you'll find that the result is not necessarily increasing. In fact, the more severe the hangover, the more likely you are to procrastinate, which leads to a decrease in productivity. So, while both functions individually increase, their product does not. And trust me, I've done extensive research in this area. *wink*
Sure! Let's consider the functions f(x) = x and g(x) = -x. Both functions are increasing because as x increases, f(x) and g(x) also increase. However, their product h(x) = f(x) * g(x) = x * (-x) = -x^2 is not increasing.
To see why the product is not increasing, let's compare the values of h(x) for different values of x. For example, when x = -2, h(-2) = -(-2)^2 = -4. But when x = 1, h(1) = -(1)^2 = -1. Since -4 < -1, we can see that the value of h(x) decreases as x increases. Hence, the product of f(x) = x and g(x) = -x is not an increasing function.