What function family has an increasing interval and a decreasing interval?

There are many families of functions or function types that have both increasing and decreasing intervals.

One example is the parabolic functions (and functions of even powers), such as f(x)=x^2 or f(x)=x^4. Namely, f(x) = x^n, where n is an element of even natural numbers.

If we let f(x) = x^2, then f'(x)=2x, which is < 0 (i.e. f(x) is decreasing) when x<0, and f'(x) > 0 (i.e. f(x) is increasing), when x > 0.

Another example are trigonometric functions, such as f(x) = sin(x). Finding the derivative (i.e. f'(x) = cos(x)) and critical points will show this.