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# Cha8 - Chapter 8 Joy Zhou Chapter 8 8.1 8.2 8.3 8.4 8.6 8.8...

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Chapter 8 Joy Zhou 10-28-2009 Chapter 8 8.1, 8.2, 8.3, 8.4, 8.6, 8.8 8.1 (a) h ( f ( t )) = | t - 1 | = braceleftBigg t - 1 if t 1 1 - t if t < 1 (1) (c) h ( h ( t ) - 1) = || t | - 1 | = braceleftBigg | t - 1 | t 0 | - t - 1 | t < 0 = t - 1 t 1 1 - t 0 t < 1 - t - 1 t ≤ - 1 t + 1 - 1 < t < 0 (2) 8.3 Note: notice the difference between part (a) and part (b). If you see these types of problems in the second mid term, please pay attention not to misunderstand the problem. (a) f ( f ( x )) = a ( ax + b ) + b (3) = a 2 x + ( ab + b ) (4) (b) Different from part (a), where f ( x ) is given, here g ( g ( x )) is given and you are asked to find g ( x ). An untold trick is to guess that g ( x ) is a linear function. 1

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Suppose g ( x ) = ax + b . Our goal is to figure out the coefficients a and b . Then g ( g ( x )) can also be written in terms of a,b and x as g ( g ( x )) = a 2 x + ( ab + b ) . Now g ( g ( x )) is known as 6 x - 8, so a 2 x + ( ab + b ) = 6 x - 8. To figure out a and b , match the coefficients of the x term and the constant term: braceleftBigg a 2 = 6 ab + b = - 8 (5) 8.4 (a) f ( x ) = 1 / 2 x + 3. f ( f ( x )) = 1 / 4 x + 9 / 2. f ( f ( f ( x ))) = 1 / 8 x + 21 / 4. They are all linear equations, so their graphs are all straight lines.
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Cha8 - Chapter 8 Joy Zhou Chapter 8 8.1 8.2 8.3 8.4 8.6 8.8...

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