Is there a way to combine multiple functions into one by averaging them, but weighing them differently depending on a variable value?
Find the equation that combines $f(x)=x^2$ and $g(x)=x$ in an average such that $f(x)$ is weighted $\frac1{x^2}$ times as heavily as $g(x).$
For all intents and purposes, the "weighing function" (in this case the inverse square) is always positive, and if negative, the absolute value is used.
Any hints?
You're almost there already. For any given $x$, you want $\frac{1}{x^2}$ of the quantity $f(x)=x^2$ for every $1$ of the quantity $g(x)=x$. Then the numerator of your "average" is $\frac{1}{x^2}\cdot f(x)+1\cdot g(x)=1+x$. Your only question then is what to divide by to keep the notion of "average." In a standard average, you would divide by $2$, since you had $1$ of the function $f$ and $1$ of the function $g$. In your new weighting system, the total of the weights is not $2$, but $1+\frac{1}{x^2}$, hence your average is $\frac{1+x}{1+\frac{1}{x^2}}=\frac{x^3+x^2}{x^2+1}=(x+1)\left(1-\frac{1}{x^2+1}\right)$.