Question ed9fd

Mar 20, 2017

See below.

Explanation:

The following answer using definitions and notations like those in Calculus by James Stewart.

By definition: A function $f$ is increasing over an interval $I$ if and only if for every ${x}_{1}$ and ${x}_{2}$ in $I$ with ${x}_{1} < {x}_{2}$, $f \left({x}_{1}\right) < f \left({x}_{2}\right)$.

The interval may be open, closed or half-open.

In many calculus classes and calculus textbooks, questions are phrased to ask students to find the open intervals on which a function is increasing and the open intervals on which it is decreasing.

For example $f \left(x\right) = {x}^{2}$ is increasing on any interval that contains no negative numbers.
So $f$ is increasing on $\left[0 , 5\right]$
and on $\left[2 , 9\right)$ and so on.

A typical text question would say $f$ is increasing on $\left(0 , \infty\right)$ because they are giving the maximal open interval on which the function is increasing.

Other notation

For the open interval I have denoted $\left(0 , \infty\right)$ some will use ]0,oo[#.