The Inverse of a Function

using CalculusWithJulia
gr()

## Plots.GRBackend()

f(x)=2^x

## f (generic function with 1 method)

plot(f,0,4)

plot!([2, 2, 0], [0, f(2), f(2)])

How to solve for an inverse function?

If we view a function as a composition of many actions, then we find the inverse by composing the inverse of these actions in reverse order.

The graph of the inverse function

The graph of \(f(x)\) is a representation of all values \((x, y)\) where \(y=f(x)\). As the inverse flips around the role of \(x\) and \(y\) we have:

If \((x,y)\) is a point on the graph of \(f(x)\), then \((y,x)\) will be a point on the graph of \(f^{-1}(x)\)

f(x) = 2^x

## f (generic function with 1 method)

xs = range(0, 2, length=50)

## 0.0:0.04081632653061224:2.0

ys=f.(xs)

## 50-element Array{Float64,1}:
##  1.0               
##  1.0286957334762774
##  1.0582149120722961
##  1.088581165149745 
##  1.1198188001321776
##  1.1519528219624953
##  1.1850089531187766
##  1.2190136542044754
##  1.25399414512947  
##  1.2899784268989174
##  ⋮                 
##  3.1898075565472563
##  3.2813414240305514
##  3.3755019229792005
##  3.4723644265096736
##  3.572006470625302 
##  3.6745078162819036
##  3.7799505132344264
##  3.8884189657157195
##  4.0

plot(xs, ys, color=:blue)

plot!(ys, xs, color=:red)

f(x)=cbrt(x)

## f (generic function with 1 method)

xs=range(-2,2,length=150)

## -2.0:0.026845637583892617:2.0

ys=f.(xs)

## 150-element Array{Float64,1}:
##  -1.2599210498948732
##  -1.2542584069509184
##  -1.2485441663946044
##  -1.2427771365281286
##  -1.2369560809963789
##  -1.2310797164530598
##  -1.2251467100693285
##  -1.2191556768718537
##  -1.2131051768958987
##  -1.2069937121375889
##   ⋮                 
##   1.2131051768958987
##   1.2191556768718537
##   1.2251467100693285
##   1.2310797164530598
##   1.2369560809963789
##   1.2427771365281286
##   1.2485441663946044
##   1.2542584069509184
##   1.2599210498948732

plot(xs, ys, color=:blue, aspect_ratio=:equal, legend=false)

plot!(ys, xs, color=:red)

plot!(identity, color=:green, linestyle=:dash)

x, y = 1/2, f(1/2)

## (0.5, 0.7937005259840998)

plot!([x,y], [y,x], color=:green, linestyle=:dot)

Lines

f(x)=sqrt(x)

## f (generic function with 1 method)

c=2

## 2

tl(x)=f(c) +  1/(2*sqrt(2)) * (x -c)

## tl (generic function with 1 method)

xs = range(0, 3, length=150)

## 0.0:0.020134228187919462:3.0

ys=f.(xs)

## 150-element Array{Float64,1}:
##  0.0                
##  0.14189513095212064
##  0.20067001862719533
##  0.24576957615571215
##  0.2837902619042413 
##  0.3172871584851762 
##  0.34757066781809537
##  0.3754192287502549 
##  0.40134003725439066
##  0.42568539285636187
##  ⋮                  
##  1.6849113254105226 
##  1.6908756319388378 
##  1.6968189741019764 
##  1.7027415714254475 
##  1.7086436396300788 
##  1.7145253907236957 
##  1.7203870330899849 
##  1.7262287715746372 
##  1.7320508075688772

zs=tl.(xs)

## 150-element Array{Float64,1}:
##  0.7071067811865477
##  0.7142253058293653
##  0.7213438304721829
##  0.7284623551150005
##  0.735580879757818 
##  0.7426994044006356
##  0.7498179290434532
##  0.7569364536862708
##  0.7640549783290884
##  0.771173502971906 
##  ⋮                 
##  1.7108187558238281
##  1.7179372804666457
##  1.7250558051094633
##  1.732174329752281 
##  1.7392928543950985
##  1.746411379037916 
##  1.7535299036807337
##  1.7606484283235513
##  1.7677669529663689

plot(xs, ys, color=:blue, legend=false)

plot!(xs, zs, color=:blue) # the tangent line

plot!(ys, xs, color=:red) # the inverse function

plot!(zs, xs, color=:red) # inverse of tangent line

If the graph of \(f(x)\) has a tangent line at \((c, f(c))\) with slope \(m\), then the graph of \(f^{-1}(x)\) will have a tangent line at \((f(c), c)\) with slope \(1/m\).