Why can some functions be inverted while others can't?
An inverse must send each output back to exactly one input, so the original function has to be bijective — one-one (no two inputs share an output) and onto (every target value is reached).
An inverse must send each output back to exactly one input, so the original function has to be bijective — one-one (no two inputs share an output) and onto (every target value is reached). If a function is many-one, like f(x) = x² on all real numbers (both +2 and −2 give 4), you can't uniquely reverse 4 back to a single input, so no inverse exists. Restrict the domain (say x ≥ 0) to make it one-one, and then it becomes invertible.
Key point
A function is invertible iff it is bijective (one-one and onto); restrict the domain to fix a many-one function.
Common mistake
assuming every function automatically has an inverse.
Memory tip
no inverse unless one-one AND onto — restrict the domain to force one-one.
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