Warm-up ideas

In Mapping a function, we explored mapping diagrams for linear functions. What happens if we now compose two functions?

  • How could we draw a mapping diagram to show the composition of the function \(f(x)=\frac{3}{2}x\) with the function \(g(x)=2x\) to get the function \(h(x)=g(f(x))\)?

We can represent this composite by using three numberlines, with arrows from the left to middle one representing \(f(x)\), and then arrows from the middle to the right representing \(g(x)\), like this:

three vertical numberlines, the left hand one from minus 1 to 5 with five arrows beginning on the left numberline starting from between 1 and 3; arrows between the numberlines as described in the text

Here, we have centred the left-hand numberline on \(2\); the numberlines are centred in such a way that the arrow starting from \(2\) is horizontal, as before.

We can then show the composite function \(h(x)=g(f(x))\) by joining the starts to the ends of the arrow chains, like this:

the same as the previous image, but now with extra arrows directly joining the left to right numberline, for example from 1 on the left to 3 on the right

This shows how the function \(h\) takes values from the left-hand numberline to values on the right-hand numberline.

As in Mapping a function, we can also extend the arrows beyond the numberlines. For clarity, we only do this for the composite function \(h\):

the same as the previous image, but now with the composite function arrows extended to lines, which meet at a point to the left of the left numberline

The lines all meet in a point, showing that \(h(x)\) is a scaling with a scale factor of \(3\). (This is the scaling factor from the left-hand to right-hand numberline.) As \(h(x)=g(f(x))=g\bigl(\frac{3}{2}x\bigr)=3x\), this is to be expected.

We can also check that the formula we worked out in Mapping a function for the location of the focal point gives us the expected location: that formula gives the location as \(\frac{1}{1-3}=-\frac{1}{2}\) times the distance between the numberlines. This matches what we see in the diagram, because the numberlines of interest here are the left-hand and right-hand ones.

  • We described linear functions as scalings. How are the scalings represented by the three functions \(f\), \(g\) and \(h\) related to each other?

\(f(x)=\frac{3}{2}x\) is a scaling with scale factor \(\frac{3}{2}\) and \(g(x)=2x\) is a scaling with scale factor \(2\). The composite function \(h(x)=3x\) is a scaling with scale factor \(3\).

In general, if we compose a scaling with scale factor \(s\) together with a scaling with scale factor \(t\), we will get a scaling with scale factor \(st\). In our particular case, we have \(\frac{3}{2}\times 2=3\).

Here, we used linear functions with no constant term and centred our mapping diagram on \(0\). Would we have obtained the same results if we had started with the functions \(f(x)=\frac{3}{2}x+4\) and \(g(x)=2x-3\), still with \(h(x)=g(f(x))\), and centred our mapping diagram on \(7\) (as input value)?