First slide

Functions (XII)

Question

A function f from integers to integers is defined as $f (n) = \{\begin{cases} n + 3, n \in odd \\ n / 2, n \in even \end{cases}$ $Suppose k \in odd and f (f (f (k))) = 27 .$ Then the value of k is ______

Moderate

A

105
B

100
C

102
D

110

Solution

$\begin{array}{l} k \in odd \\ f (k) = k + 3 \\ f (f (k)) = \frac{k + 3}{2} \end{array}$
if $\frac{k + 3}{2}$ is odd, then $27 = \frac{k + 3}{2} + 3$ or k = 45 which is not possible.
Therefore, $\frac{k + 3}{2}$ is even. Thus,
$27 = f (f (f (k))) = f (\frac{k + 3}{2}) = \frac{k + 3}{4} ∴ k = 105$
$\begin{array}{l} Verifying f (f (f (105))) = f (f (108)) = f (54) = 27 \\ ∴ k = 105 \end{array}$

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