Final Exam (with solutions)

Final Exam (with solutions) - Solutions to Math 201 Final...

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Unformatted text preview: Solutions to Math 201 Final Exam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( x- 1)- 1 3 x = 4 both sides 6: 6 1 2 ( x- 1)- 6 1 3 x = 6 4 multiply numbers to cancel denominators: 3( x- 1)- 2 x = 24 distribute: 3 x- 3- 2 x = 24 combine like terms: x- 3 = 24 both sides +3: x = 27 . 2. given equation: x 2 = 7 x- 5 2 both sides 2x: 2 x x 2 = 2 x 7 x- 2 x 5 2 cancel denominators: 6 2 x x 6 2 = 2 6 x 7 6 x-6 2 x 5 6 2 clean up: x 2 = 14- 5 x both sides- 14 and +5 x : x 2 + 5 x- 14 = 0 factor: ( x + 7)( x- 2) = 0 answers: x =- 7 or x = 2. So the sum of the two answers is- 7 + 2 = 5 . 3. Note that this equation can be written in the form 2 u 2 + 3 u = 2, where the u represents the quantity x- 1 . This is sometimes called a quadratic form equation. The strategy for solving them is to solve the equation for u first, and then when the u-values are known, solve for x from there: equation: 2 u 2 + 3 u = 2 both sides- 2: 2 u 2 + 3 u- 2 = 0 factor: (2 u- 1)( u + 2) = 0 finish solving for u : 2 u- 1 = 0 or u + 2 = 0 = u = 1 2 or u =- 2. Replace u by x- 1 : x- 1 = 1 2 or x- 1 =- 2 rewrite negative exponent: 1 x = 1 2 or 1 x =- 2 1 reciprocate (flip) both sides of each equation: x = 2 or x =- 1 2 . Note: reciprocating both sides of an equation is only valid when you have (single fraction) = (single fraction). 4. This problem is just asking you to determine the smallest and largest allowable values that h can have, based on the given inequality. If we solve the inequality, the answers are revealed: inequality: 4 h- 50 30 rewrite without the :- 30 4 h- 50 30 all sides + 50: 20 4 h 80 all sides 4: 5 h 20 . So minimum h value is 5 and maximum is 20 . 6. a) ( f + g )( x ) = f ( x ) + g ( x ) = x + 1 + x . b) ( fg )( x ) = f ( x ) g ( x ) = ( x + 1) p ( x ) . c) ( f g )( x ) = f ( g ( x )) = f ( x ) = x + 1 . d) ( g f )( x ) = g ( f ( x )) = g ( x + 1) = x + 1 . e) From part (d), ( g f )( x ) = x + 1 . The only restriction on x in this expression is caused by the squareroot. For a squareroot to be defined, the inside expression must be 0. So the domain here is x + 1 0, which is the same as x - 1, which is written as the interval- 1 , ) . 7. To find the formula for f- 1 ( x ) when you are given the formula for f ( x ): 1) write the relation down with y as the output (rather than f ( x ), if the relation was given that way); 2) exchange x and y in the relation; 3) solve the new relation for y ; 4) replace y by f- 1 ( x ). Here goes: original function: f ( x ) = 2 x- 1 put in the y : y = 2 x- 1 exchange x and y : x = 2 y- 1 solve for y : both sides + 1: x + 1 = 2 y both sides 2: x + 1 2 = y replace y by f- 1 ( x ): f- 1 ( x ) = x + 1 2 . 5. This type of function is referred to in some texts as a case-defined function. The graph of such a function is madeThis type of function is referred to in some texts as a case-defined function....
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Final Exam (with solutions) - Solutions to Math 201 Final...

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