Certainly! Let's carefully examine the given mappings to identify the underlying rule.
We have the following input-output pairs:
[tex]\[
\begin{array}{cccccc}
0 & 1 & 2 & 3 & 4 & 5 \\
\downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\
2 & 5 & 8 & 11 & 14 & 17
\end{array}
\][/tex]
To find the rule, we will analyze the relationship between each input [tex]\( x \)[/tex] and its corresponding output [tex]\( y \)[/tex].
1. Identify the change in mapping:
For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]
For [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
For [tex]\( x = 2 \)[/tex], [tex]\( y = 8 \)[/tex]
For [tex]\( x = 3 \)[/tex], [tex]\( y = 11 \)[/tex]
For [tex]\( x = 4 \)[/tex], [tex]\( y = 14 \)[/tex]
For [tex]\( x = 5 \)[/tex], [tex]\( y = 17 \)[/tex]
2. Examine the pattern:
Let's analyze the changes in the output values:
- [tex]\( 5 - 2 = 3 \)[/tex]
- [tex]\( 8 - 5 = 3 \)[/tex]
- [tex]\( 11 - 8 = 3 \)[/tex]
- [tex]\( 14 - 11 = 3 \)[/tex]
- [tex]\( 17 - 14 = 3 \)[/tex]
We see a consistent change of [tex]\( 3 \)[/tex] in the output when the input increases by [tex]\( 1 \)[/tex].
3. Determine the relationship:
Given the pattern, we can hypothesize that the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is linear: [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
From the pattern above, the slope [tex]\( m \)[/tex] is [tex]\( 3 \)[/tex] because the output increases by 3 for each increase of 1 in the input.
Let's determine the y-intercept [tex]\( c \)[/tex]:
Using the first pair [tex]\((0, 2)\)[/tex]:
[tex]\[ y = 3x + c \][/tex]
[tex]\[ 2 = 3(0) + c \][/tex]
[tex]\[ c = 2 \][/tex]
Thus, the rule is established as:
[tex]\[ y = 3x + 2 \][/tex]
4. Verify the rule:
To ensure our derived rule [tex]\( y = 3x + 2 \)[/tex] is correct, let's apply it to all given input values:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 3(0) + 2 = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 3(1) + 2 = 5 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 3(2) + 2 = 8 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 3(3) + 2 = 11 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 3(4) + 2 = 14 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 3(5) + 2 = 17 \)[/tex]
All the values match the given mapping perfectly.
Therefore, the rule for the given mapping is:
[tex]\[ y = 3x + 2 \][/tex]
