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Unformatted text preview: Assignment #10 CS 170A, Spring 2006 Due Friday, April 7 Part 1 Read chapter 11 in the text. Part 2 (to turn in) Recall that a line in the plane can be described by the two quantities: (1) the slope m , and (2) the y intersept b of the line. Indeed, points on the line are of the form ( x, f ( x )), where f ( x ) = mx + b. Any function of this form is called an affine function (contrary to popular belief, it is not a linear function in the strict mathematical sense unless b = 0). Affine functions enjoy some nice properties. First, the composition of two affine functions is again affine. In fact, we have if f 1 ( x ) = m 1 x + b 1 and f 2 ( x ) = m 2 x + b 2 , then f 1 ( f 2 ( x )) = m 1 m 2 x + ( m 1 b 2 + b 1 ); (1) i.e., the slope is m 1 m 2 and the y intercept is m 1 b 2 + b 1 . Second, the inverse of an affined function is also affine: if f ( x ) = mx + b, then f- 1 ( x ) = 1 m x- b m . (2) Third, any multiple of an affine function is affine: for any real number r , if f ( x ) = mx + b, then rf ( x ) = rmx + rb. (3) Fourth, any translation of an affine function again affine: for any real number t , we have if f ( x ) = mx + b, then f ( x ) + t = mx + ( b + t ) . (4) Fifth, the sum of two affine functions is affine: if f 1 ( x ) = m 1 x + b 1 and f 2 ( x ) = m 2 x + b 2 , then f 1 ( x ) + f 2 ( x ) = (...
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- Spring '06