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Worked solution

A periodic piecewise function

It is given that

g(x)={3,0<x≤2,5−x,2<x≤4,g(x)=\begin{cases}3,&0<x\leq2,\\5-x,&2<x\leq4,\end{cases}

and that g(x)=g(x+4)g(x)=g(x+4) for all real values of xx.

(i) Find g(3)g(3), g(8)g(8) and g(101)g(101).

(ii) Sketch the graph of y=g(x)y=g(x) for −2<x≤8-2<x\leq8.

CJC · Functions · Discussion 5 · 2025

Answer

g(3)=2g(3)=2, g(8)=1g(8)=1, g(101)=3g(101)=3.

(i)

g(3)=5−3=2g(3)=5-3=2. By periodicity, g(8)=g(4)=5−4=1g(8)=g(4)=5-4=1. Also g(101)=g(97)=g(93)=⋯=g(1)=3g(101)=g(97)=g(93)=\cdots=g(1)=3.

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(ii)

For −2<x≤0, the graph falls from the excluded point (−2,3) to the included point (0,1). On 0<x≤2 it is horizontal at 3, open at (0,3) and closed at (2,3). It falls from there to the closed point (4,1), with an open upper point (4,3) beginning the next plateau. On 4<x≤6 it is horizontal at 3, then falls to the closed endpoint (8,1). Preserve each jump: at 0 and 4 the function value is 1, not 3.

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Periodic graph on −2<x≤8: an open point (−2,3) falls to closed (0,1); an open point (0,3) begins the plateau to closed (2,3), then the graph falls to closed (4,1). An open point (4,3) begins the plateau to closed (6,3), then the graph falls to closed (8,1). At x=0 and x=4 the lower value 1 is included and the upper value 3 is excluded.
Periodic graph on −2<x≤8: an open point (−2,3) falls to closed (0,1); an open point (0,3) begins the plateau to closed (2,3), then the graph falls to closed (4,1). An open point (4,3) begins the plateau to closed (6,3), then the graph falls to closed (8,1). At x=0 and x=4 the lower value 1 is included and the upper value 3 is excluded.
Periodic graph on −2<x≤8: an open point (−2,3) falls to closed (0,1); an open point (0,3) begins the plateau to closed (2,3), then the graph falls to closed (4,1). An open point (4,3) begins the plateau to closed (6,3), then the graph falls to closed (8,1). At x=0 and x=4 the lower value 1 is included and the upper value 3 is excluded.

Common mistakes

A multiple of four maps to input 4, not to an undefined initial input 0.

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