I saw an exam question using this notation:

$$x^2\,f(x) = x^2 + 7x + 3\,.$$

I didn't understand it, so I checked the answer and I saw that was equal to:

$$f(x) = \frac{x^2 + 7x + 3}{x^2}\,.$$

Is the former a common function notation? Where or how can I find more information about it?

Thanks in advance.

  • $\begingroup$ I don't think it's a particularly special notation. It just means that $x$ can be $0$, so when dividing, you must say it's only true for $x\neq 0$ $\endgroup$
    – mrnovice
    Mar 21, 2017 at 15:20
  • 3
    $\begingroup$ Looks simply like $f(x)$ multiplied by $x^2$. $\endgroup$
    – mvw
    Mar 21, 2017 at 15:23
  • $\begingroup$ For a given $x$, $f(x)$ is a number (or something a bit more exotic, depending on the codomain of the function $f$). So $x^2f(x)$ is $x^2$, multiplied by this number: $x^2f(x) = f(x)x^2=x^2\cdot f(x)$. $\endgroup$
    – Clement C.
    Mar 21, 2017 at 15:27
  • 1
    $\begingroup$ Hahah, you are right, I can't believe it was something so simple. I think I need to sleep. Thanks! $\endgroup$
    – Emerson
    Mar 21, 2017 at 15:46

1 Answer 1


Assuming you have copied it verbatim, it's a simple redistribution of terms. Suppose for example that you wanted to subtract 3 from both sides. You would then have $$x^2 f(x) - 3 = x^2 + 7x.$$

That was just for the sake of example. Let's go back to $$x^2 f(x) = x^2 + 7x + 3.$$ Now we want to divide both sides by $x^2$. Then we have $$\frac{x^2 f(x)}{x^2} = \frac{x^2 + 7x + 3}{x^2}$$ $$\frac{x^2}{x^2} f(x) = \frac{x^2 + 7x + 3}{x^2}.$$ As we're assuming $x \neq 0$, it follows that $$\frac{x^2}{x^2} = 1,$$ and so $$1 f(x) = \frac{x^2 + 7x + 3}{x^2}$$ $$f(x) = \frac{x^2 + 7x + 3}{x^2}.$$


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