2530.41 – Composition with a Defined Operation

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

$\begin{aligned}6 \odot &= 20 \\2 \odot &= 4 \\10 \odot &= 36\end{aligned}$

and

$\begin{aligned}5 \oslash &= 4.5 \\10 \oslash &= 7 \\8 \oslash &= 6\end{aligned}$

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

Solution

From

$\begin{aligned}6 \odot &= 20 \\2 \odot &= 4 \\10 \odot &= 36\end{aligned}$

we guess that $x \odot = 4 x - 4$ . Next from

$\begin{aligned}5 \oslash &= 4.5 \\10 \oslash &= 7 \\8 \oslash &= 6\end{aligned}$

we guess that $x \oslash = \dfrac{x}{2} + 2$ . So,

$3 \odot \odot \oslash = 8 \odot \oslash = 28 \oslash = 16$