Question

Let \(S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Define \(f : S → S\) as
\(f(n) =  \begin{cases}     2n, & \text{if } n \in \{1,2,3,4,5\} \\     2n-11, & \text{if } n \in \{6,7,8,9,10\} \end{cases}\)
Let \(g : S → S\) be a function such that.
\((f \circ g)(n) =  \begin{cases}     n + 1, & \text{if } n \text{ is odd} \\     n - 1, & \text{if } n \text{ is even} \end{cases}\) 
Then \(g(10) \cdot (g(1) + g(2) + g(3) + g(4) + g(5))\) is equal to __________.

Answer 1

Correct Answer: 190

\(∵\)\(f(n) =  \begin{cases}     2n, & \text{if } n \in \{1,2,3,4,5\} \\     2n-11, & \text{if } n \in \{6,7,8,9,10\} \end{cases}\)
\(∴ f(1) = 2, f(2) = 4, … , f(5) = 10\)
and \(f(6) = 1, f(7) = 3, f(8) = 5, … , f(10) = 9\)
Now,
\(f(g(n)) =  \begin{cases}     n + 1, & \text{if } n \text{ is odd} \\     n - 1, & \text{if } n \text{ is even} \end{cases}\)
\(∴ f(g(10)) = 9 \implies g(10) = 10\)
\(f(g(1)) = 2 \implies g(1) = 1\)
\(f(g(2)) = 1 \implies g(2) = 6\)
\(f(g(3)) = 4 \implies g(3) = 2\)
\(f(g(4)) = 3 \implies g(4) = 7\)
\(f(g(5)) = 6 \implies g(5) = 3\)
\(∴\) \(g(10) \cdot (g(1) + g(2) + g(3) + g(4) + g(5)) = 190\)