Many ways problem

## Main problem

Can you arrange functions in these $2 \times 2$ tables?

Once you have completed a function square, can you find other functions which also fit?

We have provided some functions on printed cards and you can either work with the equations of the functions or with their graphs. (Please note that we haven’t excluded any input values for $x$ on the cards because one of the properties to be considered is whether these functions are defined for all real values of $x.$)

 Graph passes through $(0,0)$ Graph has rotational symmetry about $(0,0)$ $f(x)= \cdots$ Graph passes through $(0,0)$ Graph does not have rotational symmetry about $(0,0)$ $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph has rotational symmetry about $(0,0)$ $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph does not have rotational symmetry about $(0,0)$ $f(x)= \cdots$
 Function is defined for all real $x$ Different values of $x$ always give different values of $f(x)$ $f(x)= \cdots$ Function is defined for all real $x$ Different values of $x$ may give the same value of $f(x)$ $f(x)= \cdots$ Function is not defined for some real $x$ Different values of $x$ always give different values of $f(x)$ $f(x)= \cdots$ Function is not defined for some real $x$ Different values of $x$ may give the same value of $f(x)$ $f(x)= \cdots$

Could you try to form a function square for other properties of functions?

Take a look at some of the functions you have used and at any other functions you have thought about. What is the same and what is different about these functions?

From a completed function square, can you work out which properties might have been used?

Are there properties that can never be combined to make a function square?