Food for thought

## Problem

In the Warm-up we scaled (or stretched) a function $f(x)$ so that $\int_{-1}^{2} a\,f(x) \, dx = 1.$

• Which of the following functions could we scale to satisfy this condition and what is the value of $a$ in each case?

• Which ones lead to problems and why?

• How would your answers be different if we changed the condition to $\textit{an area of }\,1 \textit{ is enclosed by the curve }y=a\,f(x) \textit{, the }x\textit{-axis and the lines }x=-1\textit{ and }x=2\text{ ?}$

Can you think of some other functions which cannot be scaled to satisfy one or both of our conditions?