Building blocks

## Things you might have noticed

For some of the following functions, work out the values of $f(1)$, $f(-1)$, $f(2)$, $f(-2)$, and so on.

• $f(x)=x^2$

• $f(x)=2$

• $f(x)=1-x^2$

• $f(x)=x^4+2$

• $f(x)=1-\dfrac{1}{x^2}$

• $f(x)=x^2-2x+2$

What did you notice as you worked out these values?

Now carefully sketch the graphs of the functions on the same set of axes.

The condition that $f(-x)=f(x)$ seems to connect quite nicely to the graph $y=f(x)$ having the $y$-axis as a line of symmetry. The condition is saying that whatever our input $x$ is, if we reflect the point $(x,f(x))$ in the $y$-axis we get another point on the graph $y=f(x).$ Therefore the function behaves the same way on either side of the $y$-axis.

On a new set of axes, carefully sketch graphs of the following functions. Again, for some of the functions you may find it helpful to work out $f(1)$, $f(-1)$, $f(2)$, $f(-2)$ and so on. Note down your ideas as you sketch each graph.

• $f(x)=x$

• $f(x)=-\dfrac{1}{x}$

• $f(x)=x^3$

• $f(x)=\dfrac{x^3}{10}$

• $f(x)=x(x-1)(x+1)$

• $f(x)=1+\dfrac{1}{x}$

What did you notice this time?

We can interpret the condition $f(-x)=-f(x)$ as saying that if you reflect the point $(x, f(x))$ in the $y$-axis and then in the $x$-axis you get another point on the graph $y=f(x)$. You may be wondering how this combination of reflections corresponds to the graph $y=f(x)$ having rotational symmetry. This resource about Symmetry explores this idea.

We’ve asked you to suggest examples of functions which are even, not even, odd, or not odd. Do you think there is a function which is both even and odd?