Please excuse me if my mathematical terminology isn't exactly precise or most eloquently worded - I'm an A Level student and this is my first question.
If we have a function, $f(x)$, can we define another function, $g(x)$, whose tangent at any value for $g(x)$ is normal to $f(x)$ within the same range for $x$?
Yes. The tangent vector to $f(x)$ is $[1,\frac{\partial}{\partial x} f(x)]$, and to $g(x)$ it is $[1,\frac{\partial}{\partial x} g(x)]$. You want these vectors to be perpendicular $$[1,\frac{\partial}{\partial x} f(x)]\cdot [1,\frac{\partial}{\partial x} g(x)]=0$$ so $$1+\left(\frac{\partial}{\partial x} f(x)\right)\left(\frac{\partial}{\partial x} g(x)\right)=0$$ $$\frac{\partial}{\partial x} g(x)=\frac{-1}{\frac{\partial}{\partial x} f(x)}$$ which defines $g(x)$ up to a constant.