1.1 Review of functions

In [link] , we can see that $\left(f\circ g\right)\left(x\right)\ne \left(g\circ f\right)\left(x\right).$ This tells us, in general terms, that the order in which we compose functions matters.

Symmetry of functions

The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function $f(x)=x^4-2x^2-3$ shown in [link] (a). If we take the part of the curve that lies to the right of the y -axis and flip it over the y -axis, it lays exactly on top of the curve to the left of the y -axis. In this case, we say the function has symmetry about the y -axis    . On the other hand, consider the function $f(x)=x^3-4x$ shown in [link] (b). If we take the graph and rotate it $180\text{°}$ about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin    .

If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function $f$ has symmetry? Looking at [link] again, we see that since $f$ is symmetric about the $y$ -axis, if the point $(x,y)$ is on the graph, the point $(\text{−}x,y)$ is on the graph. In other words, $f\left(\text{−}x\right)=f\left(x\right).$ If a function $f$ has this property, we say $f$ is an even function, which has symmetry about the y -axis. For example, $f(x)=x^2$ is even because