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Inverse functions

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

$f^{-1}$ undoes $f$ : $f^{-1}(f(x)) = x$ . It exists when $f$ is one-to-one; its graph is the mirror image of $f$ 's across the line $y = x$ .

Formulas

f^{-1}(f(x)) = x, \qquad (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}

Where it shows up in computing

Monte Carlo methods★★★★★frequentScientific computing and algorithms
Inverse transform sampling: if $U$ is uniform on $(0,1)$ , $F^{-1}(U)$ has distribution $F$ .

Where it shows up in AI

Generative models★★★★★advancedAI and machine learning
Normalizing flows are invertible networks: sample with $f$ , evaluate densities with $f^{-1}$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Logarithmic functions→Algorithm analysis and complexity★★★★★
Logarithmic functions→Derivatives of elementary functions→Antiderivatives and indefinite integrals→Symbolic computation (CAS)★★★★★
Logarithmic functions→Orders of growth→Asymptotic notation (O, o, Ω, Θ)→Order of convergence→Scientific computing★★★★★
Monte Carlo methods★★★★★
Logarithmic functions→Floating point (IEEE 754)★★★★★

ℒ AI and machine learning

Logarithmic functions→Loss function★★★★★
Logarithmic functions→Loss function→Gradient descent★★★★★
Logarithmic functions→Maximum likelihood estimation→Logistic regression★★★★★
Logarithmic functions→Derivatives of elementary functions→Activation functions★★★★★
Logarithmic functions→Loss function→Regularization★★★★★
Logarithmic functions→Loss function→Support vector machines★★★★★
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⚙ Robotics and control

Logarithmic functions→Derivatives of elementary functions→Antiderivatives and indefinite integrals→Ordinary differential equations→Control theory★★★★★

⇄ Optimization and systems

Logarithmic functions→Loss function→Gradient descent→Recommender systems★★★★★

3D Computer graphics

Monte Carlo methods→The rendering equation★★★★★

∿ Signals, media and vision

Logarithmic functions→Derivatives of elementary functions→Antiderivatives and indefinite integrals→Partial fractions→Digital filters★★★★★

What depends on it

Logarithmic functions

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.