Mapping a function

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Here are mapping diagrams representing six functions of the form \(f(x)=ax+b\) for \(x\in\mathbb{R}\).

(In these diagrams, the left-hand numberline is the input and the right-hand one is the output, for example in Diagram 1, \(f(1)=3\).)

• What do you notice about these diagrams and the functions they represent? (You might find it helpful to work out some or all of the functions first.)

• What features do the diagrams and functions have in common? What are the differences between them?

Here are some features we noticed about these diagrams and functions.

• We notice that if we ignore the scales on the numberlines, all of the diagrams look identical.

• If we look at the scales, we notice that the scales on the input (left) and output (right) numberlines go up in the same step size as each other: in Diagrams 1, 2, 4 and 5, the step size is \(1\); in Diagram 6 the step size is \(0.1\), and in Diagram 3 the step size is \(0.01\).

• The middle arrow is horizontal in all of the diagrams.

• The function \(f(x)=ax+b\) represented by each diagram can be worked out either by observation or using algebra. For example, for Diagram 6:

  \[\begin{align*} f(1.0) &= 1.0a + b = 2.0\\ f(1.1) &= 1.1a + b = 2.3 \end{align*}\]

  This gives us simultaneous equations for \(a\) and \(b\); subtracting these gives \(0.1a=0.3\), so \(a=3\) and then \(b=-1\). So \(f(x)=3x-1\).

  The functions represented by each diagram are as follows:

  Diagram 1	\(f(x)=3x\)
  Diagram 2	\(f(x)=3x\)
  Diagram 3	\(f(x)=3x\)
  Diagram 4	\(f(x)=3x+1\)
  Diagram 5	\(f(x)=3x-2\)
  Diagram 6	\(f(x)=3x-1\)

• It is interesting to note that they are all of the form \(f(x)=3x+b\). The first three diagrams all representing the function \(f(x)=3x\) look the same except for their numberlines: they are centred in different places, and have different scales.

  So we wonder: Does every mapping diagram of a function of the form \(f(x)=3x+b\) look the same, no matter what the scale is? If so, can you justify this? If not, can you find an example which is different?

  We could do all sorts of things to modify the diagram, for example: draw the numberlines a different distance apart, not have the centre arrow horizonal, have different scales on the two numberlines, draw a different number of arrows, or draw the arrow start points a different distance apart. So when we think about this question, we will assume that we keep all of these things the same, and the only things we change are the choice of \(b\) and the scale on the numberlines.

  Solution

  Let’s draw a general mapping diagram for \(f(x)=3x+b\), and see what we observe. We centre the left numberline at \(c\), so the right numberline will be centred at \(3c+b\). We let the step size on the numberlines be \(s\), so the tick mark above \(c\) is at \(c+s\). This is what our mapping diagram now looks like, with just two arrows drawn for simplicity:

  

  The non-horizontal arrow starts at \(c+s\) on the input numberline, which is one tick mark above \(c\), the start of the horizontal arrow.

  On the output numberline, the non-horizontal arrow ends at \(3(c+s)+b=(3c+b)+3s\), which is \(3s\) above the horizontal arrow, represented by three tick marks on the numberline.

  So this arrow always goes from one tick mark above the centre to three tick marks above, regardless of the choice of \(b\), \(c\) or \(s\), and so this arrow will always look the same.

  We can extend this argument to other arrows: instead of starting at \(c+s\), an arrow will start at \(c+ks\), which is \(k\) tick marks above the centre, and it will end at \((3c+b)+3ks\), which is \(3k\) tick marks above the centre.

  So the mapping diagram (when centred in this way) will always look the same.