1. Prove that, for real values of \(x\), the function \[\frac{2x+1}{x^2-1}\] can take all real values.

We want to show that as we travel from \(-\infty\) to \(\infty\) along the \(x\)-axis, \(f(x)\) at some point takes each real value.

Suppose that \(y=\dfrac{2x+1}{x^2-1}\). Given a \(y\), can we find an \(x\)?

Sketch the graph of the function, and prove that it has a point of inflection between \(x=-1\) and \(x=+1\).

Things to look out for when curve sketching:

  • asymptotes;
  • intercepts;
  • stationary points and their nature;
  • behaviour as \(x\rightarrow \pm \infty\).

What is the definition of a point of inflection?