Many ways problem

## Warm-up ideas

Can you arrange the functions $f(x)=x^2$, $f(x)=-1$, $f(x)=x$ and $f(x)=x^3$ in the table below so that they have the properties stated in each cell?

 Graph is straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph is straight $y$-axis is not a line of symmetry $f(x)= \cdots$ Graph is not straight $y$-axis is a line of symmetry $f(x)= \cdots$ Graph is not straight $y$-axis is not a line of symmetry $f(x)= \cdots$

Can you explain why each function is in each cell?

We need to think about what is meant by the two properties. They both describe properties of the graph of a function, but can be interpreted algebraically too. For example, the $y$-axis is a line of symmetry if the graph would look the same when reflected in the $y$- axis, but what would this mean for the input and output values of the function?

The only way of arranging the four given functions in the table is shown here.

 Graph is straight $y$-axis is a line of symmetry $f(x)=1$ Graph is straight $y$-axis is not a line of symmetry $f(x)=x$ Graph is not straight $y$-axis is a line of symmetry $f(x)=x^2$ Graph is not straight $y$-axis is not a line of symmetry $f(x)=x^3$

There are several ways to explain why these functions have the properties in the corresponding row and column. Here are some suggestions.

• The graph of $f(x)=-1$ has gradient $0$, so it is parallel to the $x$-axis. Therefore it is straight and the $y$-axis is a line of symmetry. Another way to see the symmetry is to note that all input values of $x$ give the same output value, so in particular, $f(2)=f(-2)$, $f(3)=f(-3)$, and in general $f(x)=f(-x).$ Therefore the function behaves the same way on either side of the $y$-axis.

• The graph of $f(x)=x$ has gradient 1, so it is straight, but it does not look the same when reflected in the $y$-axis.

• The graph of $f(x)=x^2$ gets steeper and steeper as $x$-values move further away from $0$, so it is not straight, but $(-x)^2=x^2$, so the $y$-axis is a line of symmetry.

• The graph of $f(x)=x^3$ gets steeper and steeper as $x$-values move further away from $0$, so it is not straight. However, $f(-2)=(-2)^3=-8,$ and $f(2)=2^3=8$, so the graph of $f(x)=x^3$ does not look the same when reflected in the $y$-axis.

Which other functions could fit in the cells of this table?

Which straight lines are symmetric about the $y$-axis? Are all straight lines the graphs of functions of $x$?

The line $x=0$ is straight and symmetric about the $y$-axis, but a function of $x$ has to give a single output for any valid input value of $x$. If a line such as $x=3$ were the graph of a function of $x$, what could the output value, $f(3),$ be?

For the other cells, you might find it helpful to make a list of the functions you know about and try to sketch their graphs. Here is an alternative way to complete this function square.

 Graph is straight $y$-axis is a line of symmetry $f(x)=1$ Graph is straight $y$-axis is not a line of symmetry $f(x)=-x$ Graph is not straight $y$-axis is a line of symmetry $f(x)=x^2+1$ Graph is not straight $y$-axis is not a line of symmetry $f(x)=\frac{1}{x} \text{ for } x\neq 0$

We have to specify that $x$ is not zero when we write $f(x)=\dfrac{1}{x}$ because $\dfrac{1}{x}$ isn’t defined if $x=0.$ It is important to think about which inputs are valid when you write down a formula for a function.