Notation for functions

I have a basic question about the notation for writing functions. Say that I have a function defined as

$$f(x) = g(x) - 1.$$

Is it then correct to write

$$ f = g -1?$$

Are there any problems with writing functions like this?

Solution 1:

If you use $f$ as the symbol for some function, then $f(x)$ is the image of $x$ under that function, i.e. $f$ represents a function and $f(x)$ is a number, the result of $x \mapsto f(x)$. So it makes sense to write $f(x)-1$ as a function value where $1$ is subtracted from $f(x)$: there's no real risk on wrong interpretations here, this is standard notation.

However, you can also do operations on functions such as the sum of functions $f+g$ or the product of functions $fg$. In that context, you could interpret "$f-1$" as a difference of functions: the difference of $f$ and a constant function $x \mapsto 1$ (for all $x$), abbreviated simply as "$1$".

Note then that the blue $1$ in $f(x)-\color{blue}{1}$, a number, doesn't play the same role as the red $1$ in $f-\color{red}{1}$, meant to denote a function. This can raise confusion so only do this when it's sufficiently clear from the context what you mean.

Solution 2:

In $f(x) = g(x)-1$, the $1$ is a number. In $f = g - 1$, the $1$ is a function $1(x)$, the function that for any value of $x$ will always return the constant $1$. So long as you and whoever is reading your work know that difference, you should be okay to write it.