Many ways problem

## Taking it further

In the main problem you arranged four functions in a $2 \times 2$ square according to two properties that the functions did or did not have. Now we will think about arranging eight functions in a $2 \times 2 \times 2$ ‘cube’ based on whether or not the functions have three properties.

For example, can you complete this function cube for the following properties?

• Where the function is well defined, two different input values give two different output values
• The graph of function has asymptotes
• The graph of function has rotational symmetry about $(0,0)$

There are many ways to represent a cube and if you wanted to use the functions cards, you might find it simpler to arrange them in a $4 \times 2$ table, i.e. two ‘layers’ of four functions.

For example, the table below could represent a function cube for the properties

• The graph passes through $(0,0)$
• The graph is straight
• The $y$-axis is a line of symmetry for the graph
 Graph passes through $(0,0)$ Graph is straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph passes through $(0,0)$ Graph is straight $y$-axis is not a line of symmetry $f(x)= \cdots$ Graph passes through $(0,0)$ Graph is not straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph passes through $(0,0)$ Graph is not straight $y$-axis is not a line of symmetry $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph is straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph is straight $y$-axis is not a line of symmetry $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph is not straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph does not pass through $(0,0)$ Graph is not straight $y$-axis is not a line of symmetry $f(x)= \cdots$

Try to complete this function cube or make a different one.

You might need to make some extra function cards.

Make a note of any cells that you can’t think of functions for.

Make a note of properties that are difficult to combine in a function cube.