Chapter 1.6, Problem 117E

a.

To determine

To find : the encode of “call me later”.

Answer to Problem 117E

The encode for “call me later” is $\text{19 9 64 64 4 69 29 4 64 9 104 29 94}$.

Explanation of Solution

Given information :

The function for encode is $f\left(x\right)=5x+4$.

“A” for $1$ , for space $0$.

Calculation : since “A” for $1$ and for space $0$ , then for encoding “call me later” substitute value of

  $x=3,1,12,12,0,13,5,0,12,1,20,5,18$ in function,

  $\begin{array}{l}f\left(x\right)=5x+4\\ f\left(3\right)=19\text{ [substitute, multiply and add]}\\ f\left(1\right)=9\text{ [substitute, multiply and add]}\\ f\left(12\right)=64\text{ [substitute, multiply and add]}\\ f\left(0\right)=4\text{ [substitute, multiply and add]}\\ f\left(13\right)=69\text{ [substitute, multiply and add]}\\ f\left(5\right)=29\text{ [substitute, multiply and add]}\\ f\left(20\right)=104\text{ [substitute, multiply and add]}\\ f\left(18\right)=94\text{ [substitute, multiply and add]}\end{array}$

Thus, the encode for “call me later” is $\text{19 9 64 64 4 69 29 4 64 9 104 29 94}$.

b.

To determine

To find : the inverse of the function and decode a number.

Answer to Problem 117E

The inverse of the function is $f^{-1}\left(x\right)=\frac{x-4}{5}$ and the decode value is “what time”.

Explanation of Solution

Given information :

The function for encode is $f\left(x\right)=5x+4$.

“A” for $1$ , for space $0$.

Calculation : to find the inverse function substitute $f\left(x\right)=y$ ,

  $\begin{array}{l}f\left(x\right)=5x+4\\ y=5x+4\text{ [substitute }f\left(x\right)=y]\\ 5x=y-4\text{ [subtract 4 from both side]}\\ x=\frac{y-4}{5}\text{ [divide by 5]}\\ f^{-1}\left(x\right)=\frac{x-4}{5}\end{array}$

For decode substitute $x=119,44,9,104,4,104,49,69,29$ in the inverse function,

  $\begin{array}{l}{f}^{-1}\left(x\right)=\frac{x-4}{5}\\ f^{-1}\left(119\right)=23\text{ [substitute, subtract and divide]}\\ f^{-1}\left(44\right)=8\text{ [substitute, subtract and divide]}\\ f^{-1}\left(9\right)=1\text{ [substitute, subtract and divide]}\\ f^{-1}\left(104\right)=20\text{ [substitute, subtract and divide]}\\ f^{-1}\left(4\right)=0\text{ [substitute, subtract and divide]}\\ f^{-1}\left(49\right)=9\text{ [substitute, subtract and divide]}\\ f^{-1}\left(69\right)=13\text{ [substitute, subtract and divide]}\\ f^{-1}\left(29\right)=5\text{ [substitute, subtract and divide]}\end{array}$

Thus, the decode value is “what time”.