Polynomial and Rational Functions

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1 Chapter 4 Polnomial and Rational Functions 4.3 Rational Functions I 1. In R() = 4 3, the denominator, q( ) = 3, has a zero at 3. Thus, the domain of R() is all real numbers ecept 3.. In R() = 5 3 +, the denominator, q( ) = 3 +, has a zero at 3. Thus, the domain of R() is all real numbers ecept In H() = ( )( + 4), the denominator, q( ) = ( )( + 4), has zeros at and 4. Thus, the domain of H() is all real numbers ecept and In G() = ( + 3)(4 ), the denominator, q( ) = ( + 3)(4 ), has zeros at 3 and 4. Thus, the domain of G() is all real numbers ecept 3 and 4. 3( 1) 5. In F() = 5 3, the denominator, q( ) = 5 3 = ( +1)( 3), has zeros at 1 1 and 3. Thus, the domain of F() is all real numbers ecept and 3. (1 ) 6. In Q() = 3 + 5, the denominator, q( ) = = (3 1)( + ), has zeros at 1 and. Thus, the domain of Q() is all real numbers ecept 1 and In R() = 3 8, the denominator, q( ) = 3 8 = ( )( + + 4), has a zero at. ( has no real zeros.) Thus, the domain of R() is all real numbers ecept. 8. In R() = 4 1, the denominator, q( ) = 4 1 = ( 1)( + 1)( +1), has zeros at 1 and 1. ( +1 has no real zeros.) Thus, the domain of R() is all real numbers ecept 1 and In H() = , the denominator, q( ) = + 4, has no real zeros. Thus, the domain of H() is all real numbers. 40

2 Section 4.3 Rational Functions I 10. In G() = , the denominator, q( ) = 4 +1, has no real zeros. Thus, the domain of G() is all real numbers. 11. In R() = 3( 6), the denominator, q( ) = 4( 9) = 4( 3)( + 3), has zeros at 4( 9) 3 and 3. Thus, the domain of R() is all real numbers ecept 3 and In F() = ( 4) 3( ), the denominator, q( ) = 3( ) = 3( + ), has a zero at. Thus, the domain of F() is all real numbers ecept. 13. (a) Domain: { }; Range: { 1} (b) Intercept: (0, 0) (c) Horizontal Asmptote: = 1 (d) Vertical Asmptote: = (e) Oblique Asmptote: none 14. (a) Domain: { 1}; Range: { > 0} (b) Intercept: (0, ) (c) Horizontal Asmptote: = 0 (d) Vertical Asmptote: = 1 (e) Oblique Asmptote: none 15. (a) Domain: { 0}; Range: all real numbers (b) Intercepts: ( 1, 0), (1, 0) (c) Horizontal Asmptote: none (d) Vertical Asmptote: = 0 (e) Oblique Asmptote: = 16. (a) Domain: { 0}; Range: { > or < } (b) Intercepts: none (c) Horizontal Asmptote: none (d) Vertical Asmptote: = 0 (e) Oblique Asmptote: = 17. (a) Domain: {, }; Range: { 0 or > 1} (b) Intercept: (0, 0) (c) Horizontal Asmptote: = 1 (d) Vertical Asmptotes: =, = (e) Oblique Asmptote: none 18. (a) Domain: { 1, 1} ; Range: all real numbers (b) Intercept: (0, 0) (c) Horizontal Asmptote: = 0 (d) Vertical Asmptotes: = 1, = 1 (e) Oblique Asmptote: none 403

3 Chapter F( ) = + 1 Polnomial and Rational Functions 0. R() = 3+ 1 Using the function, = 1, shift the graph verticall units to up. Using the function = 1, shift the graph verticall 3 units to up. 1. R( ) = 1 ( 1) Using the function, = 1, shift the graph horizontall 1 unit to the right.. R() = 3 Using the function = 1, stretch the graph verticall b a factor of 3. = 1 3. H() = +1 Using the function = 1, shift the graph horizontall 1 unit to the left, reflect about the -ais, and stretch verticall b a factor of. 4. G() = ( + ) Using the function = 1, shift the graph horizontall units to the left, and stretch verticall b a factor of. = 1 = 404

4 Section 4.3 Rational Functions I 5. R() = = 1 ( + ) Using the function = 1, shift the graph horizontall units to the left, then reflect across the -ais 6. R() = Using the function = 1, shift the graph horizontall 1 unit to the right, and shift verticall 1 unit up. = 1 = 1 7. G() =1+ ( 3) = ( 3) +1 Using the function = 1, shift the graph 3 units right, stretch verticall b a factor of, and shift verticall 1 unit up. 8. F() = = Using the function = 1, shift the graph 1 unit left, reflect about the -ais, and shift verticall units up. = = 3 = 1 = 1 9. R() = 4 =1 4 Using the function = 1, reflect about the -ais, stretch verticall b a factor of 4 and shift verticall 1 unit up. = R() = 4 =1 4 Using the function = 1, reflect about the -ais, stretch verticall b a factor of 4 and shift verticall 1 unit up. = 1 405

5 Chapter 4 Polnomial and Rational Functions 31. R() = The degree of the numerator, p() = 3, is n =1. The degree of the denominator, q( ) = + 4, is m =1. Since n = m, the line = 3 1 = 3 is a horizontal asmptote. The denominator is zero at = 4, so = 4 is a vertical asmptote. 3. R() = The degree of the numerator, p() = 3 + 5, is n =1. The degree of the denominator, q( ) = 6, is m =1. Since n = m, the line = 3 1 = 3 is a horizontal asmptote. The denominator is zero at = 6, so = 6 is a vertical asmptote. 33. H() = The degree of the numerator, p() = , is n = 4. The degree of the denominator, q( ) = +1, is m =. Since n > m +1, there is no horizontal asmptote or oblique asmptote. The denominator has no real zeros, so there is no vertical asmptote. 34. G() = The degree of the numerator, p() = +1, is n =. The degree of the denominator, q( ) = + 5, is m = 1. Since n = m +1, there is an oblique asmptote. Dividing: ) G() = Thus, the oblique asmptote is = + 5. The denominator is zero at = 5, so = 5 is a vertical asmptote. 35. T() = The degree of the numerator, p() = 3, is n = 3. The degree of the denominator, q( ) = 4 1 is m = 4. Since n < m, the line = 0 is a horizontal asmptote. The denominator is zero at = 1 and = 1, so = 1 and = 1 are vertical asmptotes. 36. P() = The degree of the numerator, p() = 4 5, is n = 5. The degree of the denominator, q( ) = 3 1 is m = 3. Since n > m +1, there is no horizontal asmptote and there is no oblique asmptote. The denominator is zero at = 1, so = 1 is a vertical asmptote. 406

6 Section 4.3 Rational Functions I 37. Q() = The degree of the numerator, p() = 5, is n =. The degree of the denominator, q( ) = 3 4 is m = 4. Since n < m, the line = 0 is a horizontal asmptote. The denominator is zero at = 0, so = 0 is a vertical asmptote. 38. F() = = +1 ( + ) The degree of the numerator, p() = + 1, is n =. The degree of the denominator, q( ) = is m = 3. Since n < m, the line = 0 is a horizontal asmptote. The denominator is zero at = 0 and =, so = 0 and = are vertical asmptotes. 39. R() = The degree of the numerator, p() = , is n = 4. The degree of the denominator, q( ) = is m = 3. Since n = m +1, there is an oblique asmptote. Dividing: ) R() = Thus, the oblique asmptote is = 3. The denominator is zero at = 0, so = 0 is a vertical asmptote. 40. R() = = (3 + 1)( ) The degree of the numerator, p() = , is n =. The degree of the denominator, q( ) = 3 5 is m =. Since n = m, the line = 6 3 = is a horizontal asmptote. The denominator is zero at = 1 3 and =, so = 1 3 and = are vertical asmptotes. 41. G() = 3 1, 1 The degree of the numerator, p() = 3 1, is n = 3. The degree of the denominator, q( ) = is m =. Since n = m +1, there is an oblique asmptote. Dividing: 1 + ) G() = = 1 1, 1 Thus, the oblique asmptote is = G() must be in lowest terms to find the vertical asmptote: G() = 3 1 = ( 1)( + +1) ( 1) = + +1 The denominator is zero at = 0, so = 0 is a vertical asmptotes. 407

7 Chapter 4 Polnomial and Rational Functions 4. F() = 1 = 1 3 ( 1) = 1 ( 1)( +1) = 1 ( +1) The degree of the numerator, p() = 1, is n = 1. The degree of the denominator, q( ) = 3 is m = 3. Since n < m, the line = 0 is a horizontal asmptote. The denominator is zero at = 0, and = 1, so = 0, and = 1 are vertical asmptotes. 43. g( h) = (a) g( 0) = ( h) (b) g( 443) = (c) g( 8448) = m /s ( + 0) m /s ( + 443) m /s ( ) (d) (e) g( h) 14 = ( h) as h h =0 is the horizontal asmptote (f) g( h) 14 = = 0, to solve this equation would require that ( + h) = 0, which is impossible. Therefore, there is no height above sea level at which g = t 44. P( t) = ( ) ( t) (a) P( 0) = ( ) ( + 0) = 50 = 5 insects 408

8 Section 4.3 Rational Functions I (b) P( 5) = ( ( ) ( ( 5) ) = insects (c) (d) 50( t) P( t) = ( t) 50( 0.5t) = 500 as t 0.01t = 500 is the horizontal asmptote. The area can sustain a maimum population of 500 insects. (e) P( ) = ( ( ) ( ( )) A rational function R ( )has a vertical asmptote at = c whenever R c ( ) = nonzero zero. That is, whenever = c ields a zero in the denominator of the function formula. And the denominator will equal zero onl if it contains the factor c ( ) n, for some n >

(4, 2) (2, 0) Vertical shift two units downward (2, 4) (0, 2) (1, 2) ( 1, 0) Horizontal shrink each x-value is multiplied by 1 2

(4, 2) (2, 0) Vertical shift two units downward (2, 4) (0, 2) (1, 2) ( 1, 0) Horizontal shrink each x-value is multiplied by 1 2 Section. Rational Functions 9. i i i i 9i. i i i. i 7i i i i i. 9 i9 i i. g f. g f. g f, ), ), ), ), ), ), ), ), ), ), ), ) Horizontal shift two units to the right Vertical shift two units downward Vertical