 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/1/c/iii/solution

It is surjective but not injective; there is no global inverse function. Noninjectivity follows from

f (3/2) = \frac{3}{2} = f (3) .

For surjectivity, restrict to

x < 1

, where

f (x) = \frac{x}{1 - x}, f^{'} (x) = \frac{1 - x /2}{( 1 - x ) ^{3/2}} > 0.

This restriction is a continuous function, tends to

- \infty

x \to - \infty

and to

+ \infty

x \to 1^{-}

. The intermediate value theorem therefore supplies a preimage for every real number. The additional branch

x > 1

destroys global injectivity but cannot destroy surjectivity.

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