Building blocks

## Solution

We have plotted the graphs of four of the following functions.

$\sin x \quad \,\,\sin^{2}x \quad \,\,\sin(\sin x) \quad \,\, \sin (2x) \quad \,\, \arcsin x \quad \,\, \sin^{-1} x \quad \,\, \dfrac{1}{\sin x}$

Which of these functions have we plotted?

Which properties of the functions and graphs may help you to decide?

How many different functions are there in the list above?

There are many ways to decide which functions have been plotted. We will start by exploring properties of the functions and then use these properties to decide which have been plotted. You may find it helpful to try to sketch the graphs of the functions, or try to match them to the given graphs, as you read.

All of these functions are related to $\sin x$ in some way. We can think about the following questions to help identify properties of the functions.
If you find it confusing that $\sin^{2} x$ means $(\sin x)^{2}$ but $\sin^{-1} x$ does not mean $\dfrac{1}{\sin x}$, then you are in good company. The famous mathematician Carl Friedrich Gauss wrote
$\sin^{2} \varphi$ is odious to me, even though Laplace made use of it; should it be feared that $\sin \varphi^{2}$ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of $\sin(\varphi^{2})$, well then, let us write $(\sin \varphi)^{2}$, but not $\sin^{2} \varphi$, which by analogy should signify $\sin (\sin \varphi).$