[Verse 1]
Functions map from set A to set B
But can we trace our footsteps back?
Only when each element's destiny
Is one-to-one without a crack
Bijection holds the golden key
Injective meets surjective track
[Chorus]
Inverse exists when f is both ways clean
One-to-one and onto the scene
f inverse of f equals identity A
f of f inverse identity B
Perfect mirror, perfect symmetry
That's the inverse function guarantee
[Verse 2]
Left inverse means injection's true
g composed with f gives us A
Right inverse shows surjection's due
f composed with g returns to B
But when you have them both in view
They must be equal, can't you see?
[Chorus]
Inverse exists when f is both ways clean
One-to-one and onto the scene
f inverse of f equals identity A
f of f inverse identity B
Perfect mirror, perfect symmetry
That's the inverse function guarantee
[Bridge]
One-sided tells a different tale
Left inverse proves injection's there
Right inverse makes surjection sail
But full inverse needs both to share
When composition circles back again
Domain and codomain remain
[Chorus]
Inverse exists when f is both ways clean
One-to-one and onto the scene
f inverse of f equals identity A
f of f inverse identity B
Perfect mirror, perfect symmetry
That's the inverse function guarantee
[Outro]
Bijection builds the bridge complete
Where forward and backward paths meet
Every element finds its match
No collisions, nothing to catch