Answer :
To design an input-output machine, we need to establish the relationship between the input values ([tex]\( x \)[/tex]) and the output values ([tex]\( y \)[/tex]). Based on the given data for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can derive the rule.
The table provided is:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 & 13 \\ \hline y & 4 & 10 & 16 & 22 & 28 & 34 & 40 \\ \hline \end{array} \][/tex]
1. Analyze the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Let's look at the pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 10 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 16 \)[/tex]
- When [tex]\( x = 7 \)[/tex], [tex]\( y = 22 \)[/tex]
- When [tex]\( x = 9 \)[/tex], [tex]\( y = 28 \)[/tex]
- When [tex]\( x = 11 \)[/tex], [tex]\( y = 34 \)[/tex]
- When [tex]\( x = 13 \)[/tex], [tex]\( y = 40 \)[/tex]
2. Find a pattern:
To identify the rule, we observe how [tex]\( y \)[/tex] changes with respect to [tex]\( x \)[/tex]. Let's identify the differences:
- [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex] is 4.
- [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] is 10.
- [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex] is 16.
- [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] is 22.
- [tex]\( y \)[/tex] when [tex]\( x = 9 \)[/tex] is 28.
- [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex] is 34.
- [tex]\( y \)[/tex] when [tex]\( x = 13 \)[/tex] is 40.
Notice the differences in [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is increased by 2 each time:
- [tex]\( y (x = 3) - y (x = 1) = 10 - 4 = 6 \)[/tex]
- [tex]\( y (x = 5) - y (x = 3) = 16 - 10 = 6 \)[/tex]
- [tex]\( y (x = 7) - y (x = 5) = 22 - 16 = 6 \)[/tex]
- [tex]\( y (x = 9) - y (x = 7) = 28 - 22 = 6 \)[/tex]
- [tex]\( y (x = 11) - y (x = 9) = 34 - 28 = 6 \)[/tex]
- [tex]\( y (x = 13) - y (x = 11) = 40 - 34 = 6 \)[/tex]
The output [tex]\( y \)[/tex] increases by 6 for each increase of 2 in [tex]\( x \)[/tex]. This suggests a linear relationship of the form [tex]\( y = ax + b \)[/tex].
3. Formulate the equation:
From the pattern observed, we deduce that for every increase by 1 in [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value increases by half of 6, which is 3.
Let's check if [tex]\( y = 3x + c \)[/tex] fits:
- When [tex]\( x = 1 \)[/tex]:
[tex]\( 4 = 3(1) + c \)[/tex] --> [tex]\( c = 4 - 3 = 1 \)[/tex]
Hence, the equation can be written as:
[tex]\( y = 3x + 1 \)[/tex]
4. Verify the rule:
Let's verify this rule using the given pairs:
- When [tex]\( x = 1 \)[/tex]: [tex]\( y = 3(1) + 1 = 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( y = 3(3) + 1 = 10 \)[/tex]
- When [tex]\( x = 5 \)[/tex]: [tex]\( y = 3(5) + 1 = 16 \)[/tex]
- When [tex]\( x = 7 \)[/tex]: [tex]\( y = 3(7) + 1 = 22 \)[/tex]
- When [tex]\( x = 9 \)[/tex]: [tex]\( y = 3(9) + 1 = 28 \)[/tex]
- When [tex]\( x = 11 \)[/tex]: [tex]\( y = 3(11) + 1 = 34 \)[/tex]
- When [tex]\( x = 13 \)[/tex]: [tex]\( y = 3(13) + 1 = 40 \)[/tex]
All values fit the equation perfectly.
Therefore, the rule for the input-output machine is: [tex]\( y = 3x + 1 \)[/tex].
The table provided is:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 & 13 \\ \hline y & 4 & 10 & 16 & 22 & 28 & 34 & 40 \\ \hline \end{array} \][/tex]
1. Analyze the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Let's look at the pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 10 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 16 \)[/tex]
- When [tex]\( x = 7 \)[/tex], [tex]\( y = 22 \)[/tex]
- When [tex]\( x = 9 \)[/tex], [tex]\( y = 28 \)[/tex]
- When [tex]\( x = 11 \)[/tex], [tex]\( y = 34 \)[/tex]
- When [tex]\( x = 13 \)[/tex], [tex]\( y = 40 \)[/tex]
2. Find a pattern:
To identify the rule, we observe how [tex]\( y \)[/tex] changes with respect to [tex]\( x \)[/tex]. Let's identify the differences:
- [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex] is 4.
- [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] is 10.
- [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex] is 16.
- [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] is 22.
- [tex]\( y \)[/tex] when [tex]\( x = 9 \)[/tex] is 28.
- [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex] is 34.
- [tex]\( y \)[/tex] when [tex]\( x = 13 \)[/tex] is 40.
Notice the differences in [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is increased by 2 each time:
- [tex]\( y (x = 3) - y (x = 1) = 10 - 4 = 6 \)[/tex]
- [tex]\( y (x = 5) - y (x = 3) = 16 - 10 = 6 \)[/tex]
- [tex]\( y (x = 7) - y (x = 5) = 22 - 16 = 6 \)[/tex]
- [tex]\( y (x = 9) - y (x = 7) = 28 - 22 = 6 \)[/tex]
- [tex]\( y (x = 11) - y (x = 9) = 34 - 28 = 6 \)[/tex]
- [tex]\( y (x = 13) - y (x = 11) = 40 - 34 = 6 \)[/tex]
The output [tex]\( y \)[/tex] increases by 6 for each increase of 2 in [tex]\( x \)[/tex]. This suggests a linear relationship of the form [tex]\( y = ax + b \)[/tex].
3. Formulate the equation:
From the pattern observed, we deduce that for every increase by 1 in [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value increases by half of 6, which is 3.
Let's check if [tex]\( y = 3x + c \)[/tex] fits:
- When [tex]\( x = 1 \)[/tex]:
[tex]\( 4 = 3(1) + c \)[/tex] --> [tex]\( c = 4 - 3 = 1 \)[/tex]
Hence, the equation can be written as:
[tex]\( y = 3x + 1 \)[/tex]
4. Verify the rule:
Let's verify this rule using the given pairs:
- When [tex]\( x = 1 \)[/tex]: [tex]\( y = 3(1) + 1 = 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( y = 3(3) + 1 = 10 \)[/tex]
- When [tex]\( x = 5 \)[/tex]: [tex]\( y = 3(5) + 1 = 16 \)[/tex]
- When [tex]\( x = 7 \)[/tex]: [tex]\( y = 3(7) + 1 = 22 \)[/tex]
- When [tex]\( x = 9 \)[/tex]: [tex]\( y = 3(9) + 1 = 28 \)[/tex]
- When [tex]\( x = 11 \)[/tex]: [tex]\( y = 3(11) + 1 = 34 \)[/tex]
- When [tex]\( x = 13 \)[/tex]: [tex]\( y = 3(13) + 1 = 40 \)[/tex]
All values fit the equation perfectly.
Therefore, the rule for the input-output machine is: [tex]\( y = 3x + 1 \)[/tex].
