Task 160

The task

The graphs of four functions, labelled (2) through (5), are shown below. Note that the domain of function (3) is $$\{-5,-4,-3,-2,-1,0,1,2\}.$$ Find the product of the labels of the functions which are invertible. [asy] size(8cm); defaultpen(linewidth(.7pt)+fontsize(8pt)); import graph; picture pic1,pic2,pic3,pic4; draw(pic1,(-8,0)--(8,0),Arrows(4)); draw(pic1,(0,-8)--(0,8),Arrows(4)); draw(pic2,(-8,0)--(8,0),Arrows(4)); draw(pic2,(0,-8)--(0,8),Arrows(4)); draw(pic3,(-8,0)--(8,0),Arrows(4)); draw(pic3,(0,-8)--(0,8),Arrows(4)); draw(pic4,(-8,0)--(8,0),Arrows(4)); draw(pic4,(0,-8)--(0,8),Arrows(4)); real f(real x) {return x^2-2x;} real h(real x) {return -atan(x);} real k(real x) {return 4/x;} real x; draw(pic1,graph(f,-2,4),Arrows(4)); draw(pic3,graph(h,-8,8),Arrows(4)); draw(pic4,graph(k,-8,-0.125*4),Arrows(4)); draw(pic4,graph(k,0.125*4,8),Arrows(4)); dot(pic2,(-5,3)); dot(pic2,(-4,5)); dot(pic2,(-3,1)); dot(pic2,(-2,0)); dot(pic2,(-1,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(2,-2)); label(pic1,"(2)",(0,-9)); label(pic2,"(3)",(0,-9)); label(pic3,"(4)",(0,-9)); label(pic4,"(5)",(0,-9)); add(pic1); add(shift(20)*pic2); add(shift(0,-20)*pic3); add(shift(20,-20)*pic4); [/asy]