2530.41 – Composition with a Defined Operation

Let the (unary) operations ⊙\odot and ⊘\oslash be defined so that:

6⊙=202⊙=410⊙=36 \begin{aligned} 6 \odot &=& 20 \\ 2 \odot &=& 4 \\ 10 \odot &=& 36 \end{aligned}

and

5 ⊘ =4.510 ⊘ =78 ⊘ =6 \begin{aligned} 5 \oslash &=& 4.5 \\ 10 \oslash &=& 7 \\ 8 \oslash &=& 6 \end{aligned}

Guess the general rule for both operations and then find 3⊙⊙⊘.3\odot\odot\oslash.

Solution

From

6⊙=202⊙=410⊙=36 \begin{aligned} 6 \odot &=& 20 \\ 2 \odot &=& 4 \\ 10 \odot &=& 36 \end{aligned}

we guess that x⊙=4x−4x \odot = 4 x - 4.

Next from

5 ⊘ =4.510 ⊘ =78 ⊘ =6 \begin{aligned} 5 \oslash &=& 4.5 \\ 10 \oslash &=& 7 \\ 8 \oslash &=& 6 \end{aligned}

we guess that x⊘ = x/2+2.x \oslash = x/2 + 2.

So

3⊙⊙⊘=8⊙⊘=28⊘=16.3 \odot \odot \oslash = 8 \odot \oslash = 28 \oslash = 16.