Practice question · Multiple choice
A function has an inverse exactly when it is bijective. Why do both injectivity and surjectivity turn out to be necessary?
Hints
- f(x) = x squared on the reals sends 2 and -2 to 4. What should the inverse of 4 be?
- Now consider a codomain element nothing maps to. What is the inverse there?
Show the answer
B. Because an inverse must be a function on the whole codomain.
Why
The two conditions rule out the two ways an inverse fails to be a function: without injectivity the inverse of 4 must be both 2 and −2, and without surjectivity it is undefined somewhere in the codomain. Which is why restricting a domain manufactures an inverse, x² on the non-negatives is bijective, and every inverse trigonometric function is defined exactly this way.
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