2530.51 – Defined Operation 3; Do Composition

Here are two functions, hh and bb, along with some examples of what they do.

  • 3h=103h = 10

  • 7h=507h = 50

  • 5h=265h = 26


  • 4b=14b = 1

  • 7b=2.57b = 2.5

  • 20b=920b = 9

So OK then, what is nn if nhb=17.5nhb = 17.5?

(Notation: this means start with nn, do hh to it, and then do bb to that.)

Solution

Let's look at the values we've been given and see if we can find a pattern:

  • hh: 3→103 \rightarrow 10, 7→507 \rightarrow 50, 5→265 \rightarrow 26, so xh=x2+1xh = x^2 + 1.

  • bb: 4→14 \rightarrow 1, 7→2.57 \rightarrow 2.5, 20→920 \rightarrow 9, so xb=x2−1xb = \dfrac{x}{2} - 1.

So we start with the given equation for nhbnhb and use the equations above.

nhb=17.5(n2+1)b=n2+12−1=n2+12=18.5n2+1=37n2=36n=±6\begin{align*}nhb &= 17.5 \\[0.5em]\left(n^2 + 1\right)b &= \\[0.5em]\dfrac{n^2 + 1}{2} - 1 &= \\[1em]\dfrac{n^2 + 1}{2} &= 18.5 \\[1em]n^2 +1 &= 37 \\[0.5em]n^2 &= 36 \\[0.5em]n &= \pm 6\end{align*}