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Sagot :
To solve this problem, we need to construct arrow diagrams for each of the given mappings and determine whether each mapping is a one-to-one function. Let's proceed step-by-step for each mapping.
### Mapping i) [tex]\( x \rightarrow 4x + 3 \)[/tex]
1. Calculate the mappings:
- For [tex]\( x = -3 \)[/tex]: [tex]\( 4(-3) + 3 = -12 + 3 = -9 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( 4(-2) + 3 = -8 + 3 = -5 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( 4(-1) + 3 = -4 + 3 = -1 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( 4(0) + 3 = 0 + 3 = 3 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( 4(1) + 3 = 4 + 3 = 7 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 4(2) + 3 = 8 + 3 = 11 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( 4(3) + 3 = 12 + 3 = 15 \)[/tex]
2. Arrow Diagram:
[tex]\[ \begin{array}{ccccccc} -3 & \rightarrow & -9 \\ -2 & \rightarrow & -5 \\ -1 & \rightarrow & -1 \\ 0 & \rightarrow & 3 \\ 1 & \rightarrow & 7 \\ 2 & \rightarrow & 11 \\ 3 & \rightarrow & 15 \\ \end{array} \][/tex]
3. One-to-one check:
Each element in the domain [tex]\(\{-3, -2, -1, 0, 1, 2, 3\}\)[/tex] maps to a unique element in the range [tex]\(\{-9, -5, -1, 3, 7, 11, 15\}\)[/tex]. Since there are no repeated values in the range, this mapping is one-to-one.
### Mapping ii) [tex]\( x \rightarrow x^2 - 1 \)[/tex]
1. Calculate the mappings:
- For [tex]\( x = -3 \)[/tex]: [tex]\( (-3)^2 - 1 = 9 - 1 = 8 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( (-2)^2 - 1 = 4 - 1 = 3 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( (-1)^2 - 1 = 1 - 1 = 0 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( (0)^2 - 1 = 0 - 1 = -1 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( (1)^2 - 1 = 1 - 1 = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( (2)^2 - 1 = 4 - 1 = 3 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( (3)^2 - 1 = 9 - 1 = 8 \)[/tex]
2. Arrow Diagram:
[tex]\[ \begin{array}{ccccccc} -3 & \rightarrow & 8 \\ -2 & \rightarrow & 3 \\ -1 & \rightarrow & 0 \\ 0 & \rightarrow & -1 \\ 1 & \rightarrow & 0 \\ 2 & \rightarrow & 3 \\ 3 & \rightarrow & 8 \\ \end{array} \][/tex]
3. One-to-one check:
The values [tex]\(\{0, 3, 8\}\)[/tex] in the range [tex]\(\{8, 3, 0, -1, 0, 3, 8\}\)[/tex] are repeated, indicating that different elements in the domain map to the same element in the range. Specifically, [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] both map to [tex]\(8\)[/tex], [tex]\(-2\)[/tex] and [tex]\(2\)[/tex] both map to [tex]\(3\)[/tex], and [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] both map to [tex]\(0\)[/tex]. Hence, this mapping is not one-to-one.
### Summary:
- [tex]\( x \rightarrow 4x + 3 \)[/tex] is a one-to-one function.
- [tex]\( x \rightarrow x^2 - 1 \)[/tex] is not a one-to-one function.
### Mapping i) [tex]\( x \rightarrow 4x + 3 \)[/tex]
1. Calculate the mappings:
- For [tex]\( x = -3 \)[/tex]: [tex]\( 4(-3) + 3 = -12 + 3 = -9 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( 4(-2) + 3 = -8 + 3 = -5 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( 4(-1) + 3 = -4 + 3 = -1 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( 4(0) + 3 = 0 + 3 = 3 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( 4(1) + 3 = 4 + 3 = 7 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 4(2) + 3 = 8 + 3 = 11 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( 4(3) + 3 = 12 + 3 = 15 \)[/tex]
2. Arrow Diagram:
[tex]\[ \begin{array}{ccccccc} -3 & \rightarrow & -9 \\ -2 & \rightarrow & -5 \\ -1 & \rightarrow & -1 \\ 0 & \rightarrow & 3 \\ 1 & \rightarrow & 7 \\ 2 & \rightarrow & 11 \\ 3 & \rightarrow & 15 \\ \end{array} \][/tex]
3. One-to-one check:
Each element in the domain [tex]\(\{-3, -2, -1, 0, 1, 2, 3\}\)[/tex] maps to a unique element in the range [tex]\(\{-9, -5, -1, 3, 7, 11, 15\}\)[/tex]. Since there are no repeated values in the range, this mapping is one-to-one.
### Mapping ii) [tex]\( x \rightarrow x^2 - 1 \)[/tex]
1. Calculate the mappings:
- For [tex]\( x = -3 \)[/tex]: [tex]\( (-3)^2 - 1 = 9 - 1 = 8 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( (-2)^2 - 1 = 4 - 1 = 3 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( (-1)^2 - 1 = 1 - 1 = 0 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( (0)^2 - 1 = 0 - 1 = -1 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( (1)^2 - 1 = 1 - 1 = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( (2)^2 - 1 = 4 - 1 = 3 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( (3)^2 - 1 = 9 - 1 = 8 \)[/tex]
2. Arrow Diagram:
[tex]\[ \begin{array}{ccccccc} -3 & \rightarrow & 8 \\ -2 & \rightarrow & 3 \\ -1 & \rightarrow & 0 \\ 0 & \rightarrow & -1 \\ 1 & \rightarrow & 0 \\ 2 & \rightarrow & 3 \\ 3 & \rightarrow & 8 \\ \end{array} \][/tex]
3. One-to-one check:
The values [tex]\(\{0, 3, 8\}\)[/tex] in the range [tex]\(\{8, 3, 0, -1, 0, 3, 8\}\)[/tex] are repeated, indicating that different elements in the domain map to the same element in the range. Specifically, [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] both map to [tex]\(8\)[/tex], [tex]\(-2\)[/tex] and [tex]\(2\)[/tex] both map to [tex]\(3\)[/tex], and [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] both map to [tex]\(0\)[/tex]. Hence, this mapping is not one-to-one.
### Summary:
- [tex]\( x \rightarrow 4x + 3 \)[/tex] is a one-to-one function.
- [tex]\( x \rightarrow x^2 - 1 \)[/tex] is not a one-to-one function.
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