Review question

# Can we show the function $(2x+1)/(x^2-1)$ can take all real values? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9329

## Suggestion

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?