# Function notation with term before f(x)

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?

• 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 $x\neq 0$ Mar 21, 2017 at 15:20
• Looks simply like $f(x)$ multiplied by $x^2$.
• 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: $x^2f(x) = f(x)x^2=x^2\cdot f(x)$. Mar 21, 2017 at 15:27
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 $$1 f(x) = \frac{x^2 + 7x + 3}{x^2}$$ $$f(x) = \frac{x^2 + 7x + 3}{x^2}.$$