Terminology: Is $f(x)=5*g(x)$ 'increasing' in $g(x)$?

Question

Consider the following function

$$f(x)=5*g(x)$$

Can I say that this function is increasing in $g(x)$, or is the term 'increasing' only used to refer to variables, and not functions? If not, how should this relationship be described?

Answer 1

You could certainly say that $f$ is proportional to $g,$ or write $f\propto g.$ You could also say they are "ordinally equivalent," which seems to be what you're trying to say.

Two functions $f,g:X\to\Bbb R$ are said to be ordinally equivalent if one of the following (equivalent) conditions is satisfied:

• For all $x,y\in X,$ we have $f(x)\le f(y)$ iff $g(x)\le g(y).$

• There is a strictly increasing function $\phi:\Bbb R\to\Bbb R$ such that $f=\phi\circ g.$

Answer 2

This is not a function. A function needs a domain and range.

For functions you have the term of (strict) monotonicity. If your $g$ is (strict) monoton, then your $f$ is (strict) monoton.

You might check this: http://mathworld.wolfram.com/MonotonicFunction.html

Answer 3

The monotonicity is a property of the function. And $f(x)$ is in general a different function than $f(g)$. I would not use this.