Sure, let's go through each part of the question step-by-step:
### Part (a)
We are given the rule [tex]\((x, y) \rightarrow (x - 3, y + 4)\)[/tex]. We need to determine the output when the input is [tex]\((-6, 0)\)[/tex].
Follow these steps:
1. Take the [tex]\(x\)[/tex]-coordinate of the input [tex]\((-6, 0)\)[/tex], which is [tex]\(-6\)[/tex].
2. Apply the transformation to the [tex]\(x\)[/tex]-coordinate: [tex]\(x - 3\)[/tex]: [tex]\(-6 - 3 = -9\)[/tex].
3. Take the [tex]\(y\)[/tex]-coordinate of the input [tex]\((-6, 0)\)[/tex], which is [tex]\(0\)[/tex].
4. Apply the transformation to the [tex]\(y\)[/tex]-coordinate: [tex]\(y + 4\)[/tex]: [tex]\(0 + 4 = 4\)[/tex].
So, the output for the input [tex]\((-6, 0)\)[/tex] using the given rule is [tex]\((-9, 4)\)[/tex].
### Part (b)
Again, we are using the same rule [tex]\((x, y) \rightarrow (x - 3, y + 4)\)[/tex]. We need to determine the output when the input is [tex]\((3, -4)\)[/tex].
1. Take the [tex]\(x\)[/tex]-coordinate of the input [tex]\((3, -4)\)[/tex], which is [tex]\(3\)[/tex].
2. Apply the transformation to the [tex]\(x\)[/tex]-coordinate: [tex]\(x - 3\)[/tex]: [tex]\(3 - 3 = 0\)[/tex].
3. Take the [tex]\(y\)[/tex]-coordinate of the input [tex]\((3, -4)\)[/tex], which is [tex]\(-4\)[/tex].
4. Apply the transformation to the [tex]\(y\)[/tex]-coordinate: [tex]\(y + 4\)[/tex]: [tex]\(-4 + 4 = 0\)[/tex].
So, the output for the input [tex]\((3, -4)\)[/tex] using the given rule is [tex]\((0, 0)\)[/tex].
### Part (c)
To determine if the rule [tex]\((x, y) \rightarrow (x - 3, y + 4)\)[/tex] is a function, we need to check if each unique input [tex]\((x, y)\)[/tex] pair maps to a unique output [tex]\((x - 3, y + 4)\)[/tex] pair.
- For a function, each input should map to exactly one output.
- In this rule, for any given input coordinates [tex]\((x, y)\)[/tex], the output coordinates are calculated as [tex]\((x - 3, y + 4)\)[/tex].
This transformation involves basic arithmetic operations (subtraction and addition) that consistently produce one and only one output for each unique input pair. Therefore, each unique pair of input values [tex]\((x, y)\)[/tex] will generate a unique pair of output values [tex]\((x - 3, y + 4)\)[/tex].
Thus, the rule [tex]\((x, y) \rightarrow (x - 3, y + 4)\)[/tex] is indeed a function because it meets the criteria of mapping each unique input to one unique output.
### Summary Answer:
- (a) The output for input [tex]\((-6, 0)\)[/tex] is [tex]\((-9, 4)\)[/tex].
- (b) The output for input [tex]\((3, -4)\)[/tex] is [tex]\((0, 0)\)[/tex].
- (c) The rule is a function because each unique input [tex]\((x, y)\)[/tex] maps to exactly one unique output [tex]\((x - 3, y + 4)\)[/tex].
