How would you express that in terms of calculus?NWeid1 said:A function f(x) is increasing at a point x0 if and only if there exists some interval containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in the interval to the right of x0.
A monotonic function is one that either always increases or always decreases as its input increases. In other words, the function's output either consistently increases or consistently decreases as its input increases.
Yes, f(g(x)) can still be monotonic in this case. If both f and g are monotonic, then the composition of the two functions will also be monotonic. This is because the monotonicity of the composition is determined by the monotonicity of the individual functions.
If only one of the functions is monotonic, then the monotonicity of the composition will depend on the specific functions and how they interact with each other. It is possible for f(g(x)) to be monotonic, non-monotonic, or even constant in this case.
Yes, there are certain conditions that must be met for the composition to be monotonic. One important condition is that the range of g must be contained within the domain of f. Additionally, the monotonicity of f and g must be consistent in the overlapping regions of their domains.
No, f(g(x)) cannot be both increasing and decreasing at different intervals. If the composition is monotonic, it will either always increase or always decrease. This is because the monotonicity of the composition is determined by the monotonicity of the individual functions, and a function cannot be both increasing and decreasing at the same time.