Chapter 7 Rational Functions - College of the Redwoods

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702 Chapter 7 Rational Functions The notation f(x/(x + 1)) is asking us to replace each occurrence of x in the formula x/(x + 1) with the expression x/(x + 1). Confusing? Here is an easy way to think of this substitution. Suppose that we remove x from f(x) = x x + 1 , replacing each occurrence of x with empty parentheses, which will produce the template f( ) = ( ) ( ) + 1 . (15) Now, if asked to compute f(3), simply insert 3 into the blank areas between parentheses. In this case, we want to compute f(x/(x+1)), so we insert x/(x+1) in the blank space between each set of parentheses in (15) to obtain f ( ) x = x + 1 x x + 1 x x + 1 + 1 . We now have a complex fraction. The common denominator for both top and bottom of this complex fraction is x + 1. Thus, we multiply both numerator and denominator of our complex fraction by x + 1 and use the distributive property as follows. x x + 1 x x + 1 + 1 = ( ) x (x + 1) x + 1 ( ) x x + 1 + 1 (x + 1) = ( x x + 1 ( ) x (x + 1) x + 1 ) (x + 1) + (1)(x + 1) Cancel and simplify. ( ) x (x + 1) x + 1 ( ) x (x + 1) + (1)(x + 1) x + 1 = x x + (x + 1) = x 2x + 1 In the final denominator, the value x = −1/2 makes the denominator 2x + 1 equal to zero. Hence, x = −1/2 is a restriction. In the body of our work, several fractions have denominators of x + 1 and are therefore undefined at x = −1. Thus, x = −1 is a restriction. No other denominators add additional restrictions. Hence, for all values of x, except x = −1/2 and x = −1, the left-hand side of f(f(x)) = x 2x + 1 is identical to the right-hand side. Version: Fall 2007
Section 7.6 Complex Fractions 703 7.6 Exercises In Exercises 1-6, evaluate the function at the given rational number. Then use the first or second technique for simplifying complex fractions explained in the narrative to simplify your answer. In Exercises 7-46, simplify the given complex rational expression. State all restrictions. 7. 1. Given f(x) = x + 1 2 − x , evaluate and simplify f(1/2). 2. Given f(x) = 2 − x x + 5 , evaluate and simplify f(3/2). 3. Given f(x) = 2x + 3 4 − x , evaluate and simplify f(1/3). 4. Given f(x) = 3 − 2x x + 5 , evaluate and simplify f(2/5). 5. Given f(x) = 5 − 2x x + 4 , evaluate and simplify f(3/5). 6. Given f(x) = 2x − 9 11 − x , evaluate and simplify f(4/3). 8. 9. 10. 11. 5 + 6 x 25 x − 36 x 3 7 + 9 x 49 x − 81 x 3 7 x − 2 − 5 x − 7 8 x − 7 + 3 x + 8 9 x + 4 − 7 x − 9 9 x − 9 + 5 x − 4 3 + 7 x 9 x 2 − 49 x 4 17 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007
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7 Rational Functions In this chapte
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Section 7.1 Introducing Rational Fu
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Section 7.2 Reducing Rational Funct
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Section 7.3 Graphing Rational Funct
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Chapter 7 Rational Functions - College of the Redwoods

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