Chapter 3.4, Problem 47E

(a)

To determine

To whether:The encoding relation has to be a function.

Answer to Problem 47E

No, the encoding relation need not to be a function.

Explanation of Solution

The encoding does not need to be a function, because each letter could have more then one number assigned to it.

(b)

To determine

To find: The reason the graph of encoding to pass the horizontal line test.

Answer to Problem 47E

The reason the graph of encoding to pass the horizontal line test is stated below.

Explanation of Solution

The encoding will pass the horizontal test because there can only be one letter that is represented by each letter or it would be impossible to decipher the code.

Therefore, the horizontal line test is important because there is no other way to decode the function.

(c)

To determine

To find: The function that would decode the message.

Answer to Problem 47E

The function that would decode the message is $C^{-1}\left(x\right)={\left(x+2\right)}^2-3$ .

Explanation of Solution

Consider the encoded message.

  $y=-2+\sqrt{x+3}$

The decoded function is calculated as,

  $\begin{array}{c}y+2=\sqrt{x+3}\\ {\left(y+2\right)}^2=x+3\\ {\left(y+2\right)}^2-3=x\\ C^{-1}\left(x\right)={\left(x+2\right)}^2-3\end{array}$

Therefore, the decoded message is $C^{-1}\left(x\right)={\left(x+2\right)}^2-3$ .

(d)

To determine

To find: The decoding of different encoded message.

Answer to Problem 47E

The function that would decode the message is $C^{-1}\left(x\right)={\left(x+2\right)}^2-3$ .

Explanation of Solution

The decoded message is,

  $C^{-1}\left(x\right)={\left(x+2\right)}^2-3$

Encode	$C^{-1}\left(x\right)={\left(x+2\right)}^2-3$	Decode
$1$	$6$	$F$
$2.899$	$21$	$U$
$2.123$	$14$	$N$
$0.499$	$3$	$C$
$2.796$	$20$	$T$
$1.464$	$9$	$I$
$2.243$	$15$	$O$
$2.123$	$14$	$N$
$2.69$	$19$	$S$
$0$	$1$	$A$
$2.583$	$18$	$R$
$0.828$	$5$	$E$
$1$	$6$	$F$
$2.899$	$21$	$U$
$2.123$	$14$	$N$

The decoded message is tabulated in the above table.