5–24 Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of y x 2 , y x , y 1/ x , y x , or y 3 x appropriately. Then use a graphing utility to confirm that your sketch is correct. Solution: Translate left 1 unit, stretch vertically by a factor of 2, reflect over x-axis, translate down 3 units. Solution: Translate right 3 units, compress vertically by a factor of 1/2, and translate up 2 units. Solution: y = (x + 3)2 = 9; translate left 3 units and down 9 units. Solution: y = 1/2 [(x - 1)2 + 2]; translate right 1 unit and up 2 units, compress vertically by a factor of 1 Solution: Translate left 1 unit, reflect over x-axis, translate up 3 units. Solution: Translate right 4 units and up 1 unit. Solution: Compress vertically by a factor of 1/2 , translate up 1 unit. Solution: Stretch vertically by a factor of 3 and reflect over x-axis. Solution: Translate right 3 units. Solution: Translate right 1 unit and reect over x-axis. Solution: Translate left 1 unit, reflect over x-axis, translate up 2 units. Solution: y = 1-1/x; reflect over x-axis, translate up 1 unit. Solution: Translate left 2 units and down 2 units. Solution: Translate right 3 units, reflect over x-axis, translate up 1 unit. Solution: Stretch vertically by a factor of 2, translate right 1/2 unit and up 1 unit. Solution: y = |x -2|; translate right 2 units. Solution: Stretch vertically by a factor of 2, reflect over x-axis, translate up 1 unit. Solution: Translate right 2 units and down 3 units. Solution: Translate left 1 unit and up 2 units. Solution: Translate right 2 units, reflect over x-axis. 25. (a) Sketch the graph of y = x + |x| by adding the corresponding y-coordinates on the graphs of y = x and y = |x|. (b) Express the equation y = x + |x| in piecewise form with no absolute values, and confirm that the graph you obtained in part (a) is consistent with this equation. Solution: 26. Sketch the graph of y = x + (1/x) by adding corresponding y-coordinates on the graphs of y = x and y = 1/x. Use a graphing utility to confirm that your sketch is correct. Solution: Questions 27–28: Find formulas for f + g, f − g, fg, and f /g, and state the domains of the functions. x 1 , g ( x) 28. f ( x) 2 1 x x Solution: 2 x 1 x x 1 x x2 1 x2 2x2 1 f g 1 x2 x x 1 x2 x 1 x2 x 1 x2 Domain of f; (-ꝏ ,+ ꝏ) Domain of g; x≠ 0 ,or (-ꝏ,0)U(0 ,+ ꝏ) Domain f+g =Domain of f Ⴖ Domain of g Domain of f+ g = (-ꝏ ,+ ꝏ) Ⴖ (-ꝏ,0)U(0 ,+ ꝏ) Domain of f+ g = x≠ 0 or (-ꝏ,0)U(0 ,+ ꝏ) 2 x 1 x x 1 x x2 1 x2 1 f g 2 2 2 1 x x x 1 x x 1 x x 1 x2 Domain f - g =Domain of f Ⴖ Domain of g Domain of f - g = (-ꝏ ,+ ꝏ) Ⴖ (-ꝏ,0)U(0 ,+ ꝏ) Domain of f - g = x≠ 0 or (-ꝏ,0)U(0 ,+ ꝏ) 1 x 1 fg 2 2 1 x x 1 x Domain f g =Domain of f Ⴖ Domain of g Domain of f g = (-ꝏ ,+ ꝏ) Ⴖ (-ꝏ,0)U(0 ,+ ꝏ) Domain of f g = x≠ 0 or (-ꝏ,0)U(0 ,+ ꝏ) x f 1 x2 x2 1 g 1 x2 x Domain f/ g = (Domain of f Ⴖ Domain of g) \g(x)=0 Domain of f/ g =[ (-ꝏ ,+ ꝏ) Ⴖ (-ꝏ,0)U(0 ,+ ꝏ)] \1/x=0 Domain of f/ g = x≠ 0 or (-ꝏ,0)U(0 ,+ ꝏ) Solution of Q.27: Solution of Q.28: Solution of Q.29: Question 30. Let g x x . Find (a). g (5s 2) 1 (d ). g ( x) (b). g ( x 2) (c).3g (5 x) ( f ). g ( x) g ( x 2 ) 2 (e). g ( g ( x)) ( g ). g (1/ x ) (h). g x 1 Solution: We have given g x x 2 g x x (a). g (5s 2) 5s 2 (b). g ( x 2) (i). g ( x h) g x x x 2 (c).3g (5 x) 3 5 x g x x 1 1 g ( x) x g x x (d ). (e). g ( g ( x)) g ( x ) ( f ). g ( x) g ( x 2 ) 2 ( g ). g ( 1 ) x (h). g x 1 2 g x x x x 2 x2 x x 0 1 x g x x g x x x 1 g x x x 1 2 g x x (i ). g ( x h) x h Question 31–34: Find formulas for f g and g f , and state the domains of the compositions. Solution of Q.31: Solution of Q.32: f g x f ( x) x 3, f ( g ( x)) f As f ( x) x 3 f Domain of Or g x f x2 3 x2 3 x2 3 3 x2 3 3 x 2 3 3 : x2 3 3 0 x2 3 3 , Domain of g ( x) x 2 3 x 2 3 9, x 2 3 3 9 3, x 2 6, x 6 6, g : , , 6 U x 2 3 3 , , 6 U Then, Domain of f 6, , 6 U 6, g f x g ( f ( x)) g As g ( x 2 ) x 3 g f x 2 x 3 3 x 3 3 x x 3, x2 3 0, hence , x : x 0, 0, Domain of f: x 3 0, x 0, Domain of g: Domain of Then Domain of g f :, 3, 0, 3, Solution of Q.33: x 1 , g ( x) 2 1 x x 1 f g x f ( g ( x)) f x 1 1 x 1 x As f 2 2 2 x 1 x 1 x 1 x 2 1 x x x f g x 2 x 1 Solution of Q.34: f ( x) Domain of f: x2 1 0, , Domain of g: x 0, hence ,0 0, Then Domain of f g : , 0 0, , , 0 0, x f x g ( f ( x)) g 2 1 x x 1 1 x2 1 ) x As g ( x 1 x2 x x 2 1 x 1 g f x x g , x 0, hence ,0 0, Domain of f: x2 1 0, Domain of g: 1 Domain of x : x 0, , 0 0, x Then Domain of g f :, , , 0 0, , 0 0, Question 35–36: Find a formula for f ◦g◦h. Solution of Q.35: f 36: Find g h ( x) if f ( x) 1 , 1 x g ( x) 3 x , h( x ) 1 x3 Solution: We have f g h ( x) f ( g (h( x))) h( x ) 1 x3 g( ) 3 f g (h( x) ) 3 h( x) 1 1 f g ( h( x ) ) 1 1 1 x g ( h( x ) ) 3 1 1 x3 x 1 x x 1 x 1 x x x 1 Question 37–42: Express f as a composition of two functions; that is, find g and h such that f = g◦h. [Note: Each exercise has more than one solution.] Solution of Q.37: f g h ( x) f ( g (h( x))) Solution of Q.38: (a). f ( x) x2 1 h x 1, and Then g ( x) x 1, and g h g h x 1 h x x2 1 1 1 , and g h g h( x) x 3 h( x ) h( x ) 1 Then g ( x) , and h x x 3 x Solution of Q.39: (b). f ( x) Solution of Q.40: (a). f ( x) 3sin( x2 ) 3sin h( x) , and Then g ( x) 3sin x, and g h g h( x) 3sin h( x) h x x2 (b). f ( x) 3sin 2 x 4sin x 3 h( x) 4 h( x) , and 2 Then g ( x) 3x2 4 x, and Solution of Q.41: h x sin x g h g h( x ) 3 h ( x ) 4 h ( x ) 2 Solution of Q.42: (a). f ( x) 1 1 and 2 1 x 1 h( x ) Then g ( x) 1 , and 1 x (b). f ( x) 5 2 x 5 h( x) g h g h( x ) h x x2 and Then g ( x) 5 2 x , and 1 1 h( x ) g h g h( x) 5 h( x) h x 2x 63. In each part, classify the function as even, odd, or neither. Solution: