Calculus I

Lesson 18: Inverse Functions

Example 1
For each of the following functions, determine whether the

function is one-to-one. If yes, plot the function and its

inverse on the same axes.

> restart; with(plots):

Warning, the name changecoords has been redefined

1a) .

> f1:= x -> 2*x + 1;

> g1:= x -> .5 * ( x - 1);

> a1:= plot(f1(x), x = -10..10, thickness=2, color = black):

> b1:= plot(g1(x), x = -10..10, thickness=2, color = red):

> c1:= plot(x, x = -10..10, thickness=2, color = blue):

> d1:= textplot([2,10,`f1`], thickness=2, color = magenta):

> e1:= textplot([7,10,`y=x`], thickness=2, color = magenta):

> h1:= textplot([3,-3,`g1`], thickness=2, color = magenta):

> display({a1,b1,c1,d1,e1,h1});

1b)

> f2:= x -> x^3;

> g2:= x -> surd(x,3);

> a2:=plot(f2(x), x = -1.5..1.5, thickness=2, color = black):

> b2:= plot(g2(x), x = -1.5..1.5, thickness=2, color = red):

> c2:= plot(x, x = -1.5..1.5, thickness=2, color = blue):

> d2:= textplot([1.1,.5,`f2`], thickness=2, color = magenta):

> e2:= textplot([1.5,2,`y=x`], thickness=2, color = magenta):

> h2:= textplot([0.3,1.5,`g2`], thickness=2, color = magenta):

> display({a2,b2,c2,d2,e2,h2});

1 c) , x >= 2

> f3:= x -> sqrt(x - 2);

> g3:= x -> x^2 + 2;

> a3:= plot(f3(x), x = 2..4, thickness=2, color = black):

> b3:= plot(g3(x), x = 0..4, thickness=2, color = red):

> c3:= plot(x, x = 0..4, thickness=2, color = blue):

> d3:= plot([2,t,t = 0..6], thickness=2, color = brown):

> e3:= textplot([3,14,`f3`], thickness=2, color = magenta):

> h3:= textplot([3,4.3,`y=x`], thickness=2, color =magenta):

> i3:= textplot([3.5,-1.5,`g3`], thickness=2, color = magenta):

> display({a3,b3,c3,d3,e3,h3,i3});

1d) , x in [0,3]

> f4:= x -> 9 - x^2;

> g4:= x -> sqrt(9-x);

> a4:= plot(f4(x), x = -3..3, thickness=2, color = black):

> b4:= plot(g4(x), x = 0..9, thickness=2, color = red):

> c4:= plot(x, x = 0..9, thickness=2, color = blue):

> d4:= textplot([2.3,7,`f4`], thickness=2, color = magenta):

> e4:= textplot([6,5,`y=x`], thickness=2, color = magenta):

> h4:= textplot([5,1.5,`g4`], thickness=2, color = magenta):

> display({a4,b4,c4,d4,e4,h4});

1e) , x > 1

> f5:= x -> 1/ (x-1);

> g5:= x -> (1 + x) / x;

> a5:= plot(f5(x), x = 2..30, thickness=2, color = black):

> b5:= plot(g5(x), x = .03.. 1, thickness=2, color = red):

> c5:= plot( x, x = 0..10, thickness=2, color = blue):

> d5:= textplot([6,2,`f5`], thickness=2, color = magenta):

> e5:= textplot([4,6,`y=x`], thickness=2, color = magenta):

> h5:= textplot([2,10,`g5`], thickness=2, color = magenta):

> display({a5,b5,c5,d5,e5,h5});

Example 2
Plot y = sin(x) for x in [ - /2, /2] and also plot the inverse function.

> t1:= x -> sin(x);

> it1:= x -> arcsin(x);

> u1:= plot(t1(x), x = -Pi/2..Pi/2, thickness=2, color = black):

> u2:= plot(it1(x), x = -1..1, thickness=2, color = red):

> u3:= plot(x, x = -Pi/2..Pi/2, thickness=2, color = blue):

> u4:= textplot([1,.7,`sin(x)`], thickness=2, color = magenta):

> u5:= textplot([1.3,1.6,`y = x`], thickness=2, color = magenta):

> u6:= textplot([.7,1.4,`arcsin(x)`], thickness=2, color = magenta):

> display({u1,u2,u3,u4,u5,u6});

Example 3
Plot y = cos(x) for x in [ 0, ]. Also plot the inverse function.

> t2:= x -> cos(x);

> it2:= x -> arccos(x);

> v1:= plot(t2(x), x = 0..Pi, thickness=2, color = black):

> v2:= plot(it2(x), x = -1..1, thickness=2, color = red):

> v3:= plot(x, x = 0..Pi, thickness=2, color = blue):

> v4:= textplot([1.6,.6,`cos(x)`], thickness=2, color = magenta):

> v5:= textplot([2,1.6,`y=x`], thickness=2, color = magenta):

> v6:= textplot([.7,1.7,`arcos(x)`], thickness=2, color = magenta):

> display({v1,v2,v3,v4,v5,v6});

Example 4
Plot f(x) = tan(x) for x in ( - /2, /2). Also plot the inverse function.

> t3:= x -> tan(x);

> it3:= x -> arctan(x);

> w1:= plot(t3(x), x = -Pi/2 + 0.1.. Pi/2 - 0.1, thickness=2, color = black):

> w2:= plot(it3(x), x = -10..10, thickness=2, color = red):

> w3:= plot(x, x = -10..10, thickness=2, color = blue):

> w4:= textplot([5.5,2.5,`arctan(x)`], thickness=2, color = magenta):

> w5:= textplot([8.5,7,`y=x`], thickness=2, color = magenta):

> w6:= textplot([3,8,`tan(x)`], thickness=2, color = magenta):

> display({w1,w2,w3,w4,w5,w6});

Example 4
Plot f(x) = csc(x) for x in (- /2,0) and (0, /2) . Also plot the inverse function.

> t4:= x -> csc(x);

> it4:= x -> arccsc(x);

> z1:= plot(csc(x), x = -Pi/2 ..- 0.1, thickness=2, color = black):

> z2:= plot(csc(x), x = 0.1..Pi/2, thickness=2, color = black):

> z3:= plot(it4(x), x = 1..10, thickness=2, color = red):

> z4:= plot(it4(x), x = -10..-1, thickness=2, color = red):

> z5:= plot(x, x= -10..10, thickness=2, color = blue):

> z6:= textplot([1.8,5,`csc(x)`], thickness=2, color = magenta):

> z7:= textplot([5,4,`y=x`], thickness=2, color = magenta):

> z8:= textplot([-6,-2,`arccsc(x)`], thickness=2, color = magenta):

> display({z1,z2,z3,z4,z5,z6,z7,z8});