- 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?
