 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/1/e/ii/solution

The algebraic identity

x^{3} + x^{- 3} = (x + x^{- 1})^{3} - 3 (x + x^{- 1})

suggests $f (t) = t^{3} - 3 t$ for every $t > 0$ , and substitution verifies the functional equation.

x + 1/ x \geq 2

, since

(x - 1)^{2} / x \geq 0

. Conversely, for each

t \geq 2

x^{2} - t x + 1 = 0

(t + t^{2} - 4) /2

f (t) = t^{3} - 3 t

exactly on

[2, \infty)

; its values on

(0, 2)

are arbitrary. The displayed function is one convenient choice on the full required domain.

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