Given the range $(1,1),(4,2),(2,-1)$, with a coordinate transformation of $f(x, y)=(x+1, y-1)$, what is the domain?
A. $(2,0),(5,1),(3,-2)$
B. $(0,2),(3,3),(1,0)$
C. $(0,0),(3,1),(1,-2)$
D. $(1,1),(2,4),(2,-1)$
Answer (2)
Apply the transformation f ( x , y ) = ( x + 1 , y − 1 ) to each point in the given range.
Calculate the transformed points: f ( 1 , 1 ) = ( 2 , 0 ) , f ( 4 , 2 ) = ( 5 , 1 ) , and f ( 2 , − 1 ) = ( 3 , − 2 ) .
The transformed points form the domain.
The domain after the transformation is ( 2 , 0 ) , ( 5 , 1 ) , ( 3 , − 2 ) .
Explanation
Understanding the Problem We are given a range of points ( 1 , 1 ) , ( 4 , 2 ) , ( 2 , − 1 ) and a coordinate transformation f ( x , y ) = ( x + 1 , y − 1 ) . We need to find the domain after applying this transformation to the given range.
Applying the Transformation To find the domain, we apply the transformation to each point in the given range. This means we substitute the x and y values of each point into the transformation function.
Transforming the First Point Let's apply the transformation to the first point ( 1 , 1 ) .
f ( 1 , 1 ) = ( 1 + 1 , 1 − 1 ) = ( 2 , 0 )
Transforming the Second Point Now, let's apply the transformation to the second point ( 4 , 2 ) .
f ( 4 , 2 ) = ( 4 + 1 , 2 − 1 ) = ( 5 , 1 )
Transforming the Third Point Next, let's apply the transformation to the third point ( 2 , − 1 ) .
f ( 2 , − 1 ) = ( 2 + 1 , − 1 − 1 ) = ( 3 , − 2 )
Determining the Domain The transformed points are ( 2 , 0 ) , ( 5 , 1 ) , ( 3 , − 2 ) . Therefore, the domain after the transformation is ( 2 , 0 ) , ( 5 , 1 ) , ( 3 , − 2 ) .
Examples
Coordinate transformations are used extensively in computer graphics and image processing. For example, if you want to shift an image one pixel to the right and one pixel down, you would apply the transformation f ( x , y ) = ( x + 1 , y − 1 ) to each pixel in the image. This concept is also fundamental in robotics, where robot movements are planned using coordinate transformations to navigate and manipulate objects in space.
The domain after applying the transformation f ( x , y ) = ( x + 1 , y − 1 ) to the range ( 1 , 1 ) , ( 4 , 2 ) , ( 2 , − 1 ) is ( 2 , 0 ) , ( 5 , 1 ) , ( 3 , − 2 ) . Therefore, the chosen option is A.
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