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Chapter 1 Answers Homework EVEN #

Chapter 1 Answers Homework EVEN # - Chapter 1 Functions...

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1 Chapter 1 Functions, Graphs, and Limits 1.1 Functions 2. f ( x ) = 7 x + 1 f (0) = 7(0) + 1 = 1 f (1) = 7(1) + 1 = 6 f ( 2) = 7( 2) + 1 = 15 4. 3 3 3 3 ( ) (2 1) ( 1) ( 2 1) 1 (0) (0 1) 1 (1) (2 1) 27 h t t h h h = + = − + = − = + = = + = 6. 2 2 2 2 2 ( ) (2) 5 1 2 1 0 1 1 (0) 0 ( 1) 0 1 2 ( 1) 1 x f x f x f f = = = + + = = = = − + + 8. ( ) 3 2 3 2 3 2 3 3 2 ( ) ( 1) (0) (0 1) 1 ( 1) ( 1 1) 0 (8) (8 1) 9 27 g u u g g g = + = + = = − + = = + = = 10. 1 ( ) 3 2 1 (1) 1 3 2(1) 1 1 ( 3) 3 3 2( 3) 1 1 (0) 3 2(0) 3 = = = = = = = f t t f f f 12. ( ) 4 ( 2) 4 2 6 (0) 4 0 4 (2) 4 2 6 g x x g g g = + = + − = = + = = + = 14. 3 if 5 ( ) 1 if 5 5 if 5 < − = + > t f t t t t t f ( 6) = 3 f ( 5) = 5 + 1 = 4 (16) 16 4 = = f 16. Since 2 1 0 x = for 1 x = ± , f ( x ) is defined only for 1 x ≠ ± and the domain does not consist of the real numbers. 18. The square root function only makes sense for non-negative numbers. Since 2 1 0 t + for all real numbers t the domain of 2 ( ) 1 h t t = + consists of all real numbers. 20. 3 2 ( ) 3 2 5 f x x x x = + + The domain consists of all reals. 22. 2 2 1 ( ) 2 2 ( 2)( 1) 0 t f t t t t t t t + = = + if 1 and 2. t t ≠ − 24. 2 ( ) 4 h s s = is defined only if 2 4 0 s or equivalently ( 2)( 2) 0 s s + . This occurs when the factors ( 2) s and ( 2) s + are zero or have the same sign. This happens when 2 s or 2 s ≤ − and these values of s form the domain of h . 26. 2 2 2 ( ) 4 ( 1) ( 1) 4 2 5 f u u f x x x x = + = + = +

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2 Chapter 1. Functions, Graphs, and Limits 28. 2 ( ) (2 10) f u u = + [ ] 2 2 2 ( 5) 2( 5) 10 (2 10 10) 4 f x x x x = + = + = 30. 2 2 1 ( ) 1 ( 2) 2 f u u f x x x x = + = + 32. 2 2 1 1 ( ) , 1 ( 1) f u u f x x = = 34. For ( ) 2 3 f x x = + , ( ) ( ) (2( ) 3) (2 3) 2 2 3 2 3 2 2 f x h f x x h x h h x h x h h h + + + + = + + = = = 36. 2 ( ) = f x x 2 2 2 2 2 ( ) ( ) ( ) 2 (2 ) 2 + + = + + = + = = + f x h f x x h x h h x xh h x h h x h h x h 38. 1 ( ) = f x x 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) + + + = + + = + = + = + = + x h x f x h f x x x h h h x x h x x h hx x h x x h hx x h h hx x h x x h 40. 2 ( ) 1, ( ) 1 f x x g x x = + = 2 2 2 2 ( ( )) (1 ) 1 2 2 and ( ( )) 1 ( 1) f g x x x x g f x x x = + = + = + = − So ( ( )) ( ( )) f g x g f x = means 2 2 2 2 2 2 2 2 2 0 1 0 x x x x x x x + = − + = + = but, by the quadratic formula, this last equation has no solutions. 42. ( ) 4 2 1 2 ( ( )) 4 x x x f g x x + + = = , 1 1 4 4 1 ( ( )) 2 1 2 x x x g f x x = = + + so ( ( )) ( ( )) f g x g f x = means 2 4 1 4 2 1 x x x x + = + . Clearing denominators, multiplying, and collecting terms gives 2 2 2 2 2 (2 )(2 1) (4 1)(4 ) 2 5 2 4 17 4 6 12 6 6( 2 1) 6( 1) 0 x x x x x x x x x x x x x + + = + + = − + + = + = = The last equation has solution 1 x = which is in the domains of ( ( )) f g x and ( ( )) g f x . Thus ( ( )) ( ( )) f g x g f x = only for 1 x = . 44. 2 2 2 ( ) 5 ( 1) ( 1) 5 2 6 f x x f x x x x = + + = + + = + + 46. 2 ( ) (2 6) f x x = [ ] 2 2 2 ( 3) 2( 3) 6 (2 6 6) 4 f x x x x + = + = + =
Chapter 1. Functions, Graphs, and Limits 3 48. 1 2 ( ) 3 1 1 2 3 (3) 2 x f x x x f x x x x = + = + = + 50. ( ) 2 20 f x x = 2 2 2 2 ( 2 9) 2( 2 9) 20 2 4 18 20 2 4 2 f x x x x x x x x + = + = + = 52. 5 2 3 3 5 2 ( ) ( 3 12) ( ) ( ) 3 12 f x x x g u u h x x x = + = = + 54.

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Chapter 1 Answers Homework EVEN # - Chapter 1 Functions...

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