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Study Guides > MATH 1314: College Algebra

Solutions

Solutions to Try Its

1. f1(f(x))=f1(x+53)=3(x+53)5=(x5)+5=x{f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left(\frac{x+5}{3}\right)=3\left(\frac{x+5}{3}\right)-5=\left(x - 5\right)+5=x\\ and f(f1(x))=f(3x5)=(3x5)+53=3x3=xf\left({f}^{-1}\left(x\right)\right)=f\left(3x - 5\right)=\frac{\left(3x - 5\right)+5}{3}=\frac{3x}{3}=x\\ 2. f1(x)=x34{f}^{-1}\left(x\right)={x}^{3}-4\\ 3. f1(x)=x1{f}^{-1}\left(x\right)=\sqrt{x - 1}\\ 4. f1(x)=x232,x0{f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{2},x\ge 0\\ 5. f1(x)=2x+3x1{f}^{-1}\left(x\right)=\frac{2x+3}{x - 1}\\

Solutions to Odd-Numbered Exercises

1. It can be too difficult or impossible to solve for x in terms of y. 3. We will need a restriction on the domain of the answer. 5. f1(x)=x+4{f}^{-1}\left(x\right)=\sqrt{x}+4\\ 7. f1(x)=x+31{f}^{-1}\left(x\right)=\sqrt{x+3}-1\\ 9. f1(x)=x53{f}^{-1}\left(x\right)=-\sqrt{\frac{x - 5}{3}}\\ 11. f(x)=9xf\left(x\right)=\sqrt{9-x}\\ 13. f1(x)=x53{f}^{-1}\left(x\right)=\sqrt[3]{x - 5}\\ 15. f1(x)=4x3{f}^{-1}\left(x\right)=\sqrt[3]{4-x}\\ 17. f1(x)=x212,[0,){f}^{-1}\left(x\right)=\frac{{x}^{2}-1}{2},\left[0,\infty \right)\\ 19. f1(x)=(x9)2+44,[9,){f}^{-1}\left(x\right)=\frac{{\left(x - 9\right)}^{2}+4}{4},\left[9,\infty \right)\\ 21. f1(x)=(x92)3{f}^{-1}\left(x\right)={\left(\frac{x - 9}{2}\right)}^{3}\\ 23. f1(x)=28xx{f}^{-1}\left(x\right)={\frac{2 - 8x}{x}}\\ 25. f1(x)=7x31x{f}^{-1}\left(x\right)=\frac{7x - 3}{1-x}\\ 27. f1(x)=5x44x+3{f}^{-1}\left(x\right)=\frac{5x - 4}{4x+3}\\ 29. f1(x)=x+11{f}^{-1}\left(x\right)=\sqrt{x+1}-1\\ 31. f1(x)=x+6+3{f}^{-1}\left(x\right)=\sqrt{x+6}+3\\ 33. f1(x)=4x{f}^{-1}\left(x\right)=\sqrt{4-x}\\ Graph of f(x)=4- x^2 and its inverse, f^(-1)(x)= sqrt(4-x). 35. f1(x)=x+4{f}^{-1}\left(x\right)=\sqrt{x}+4\\ Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4. 37. f1(x)=1x3{f}^{-1}\left(x\right)=\sqrt[3]{1-x}\\ Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3). 39. f1(x)=x+8+3{f}^{-1}\left(x\right)=\sqrt{x+8}+3\\ Graph of f(x)= x^2-6x+1 and its inverse, f^(-1)(x)= sqrt(x+8)+3. 41. f1(x)=1x{f}^{-1}\left(x\right)=\sqrt{\frac{1}{x}}\\ Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x). 43. [2,1)[3,)\left[-2,1\right)\cup \left[3,\infty \right)\\ Graph of f(x)= sqrt((x+2)(x-3)/(x-1)). 45. [4,2)[5,)\left[-4,2\right)\cup \left[5,\infty \right)\\ Graph of f(x)= sqrt((x^2-x-20)/(x-2)). 47. (2,0);(4,2);(22,3)\left(-2, 0\right); \left(4, 2\right); \left(22, 3\right)\\ Graph of f(x)= x^3-x-2. 49. (4,0);(0,1);(10,2)\left(-4, 0\right); \left(0, 1\right); \left(10, 2\right)\\ Graph of f(x)= x^3+3x-4. 51. (3,1);(1,0);(7,1)\left(-3, -1\right); \left(1, 0\right); \left(7, 1\right)\\ Graph of f(x)= x^4+5x+1. 53. f1(x)=x+b24b2{f}^{-1}\left(x\right)=\sqrt{x+\frac{{b}^{2}}{4}}-\frac{b}{2}\\ 55. f1(x)=x3ba{f}^{-1}\left(x\right)=\frac{{x}^{3}-b}{a}\\ 57. t(h)=200h4.9t\left(h\right)=\sqrt{\frac{200-h}{4.9}}\\, 5.53 seconds 59. r(V)=3V4π3r\left(V\right)=\sqrt[3]{\frac{3V}{4\pi }}\\, 3.63 feet 61. n(C)=100C25.6Cn\left(C\right)=\frac{100C - 25}{.6-C}\\, 250 mL 63. r(V)=V6πr\left(V\right)=\sqrt{\frac{V}{6\pi }}\\, 3.99 meters 65. r(V)=V4πr\left(V\right)=\sqrt{\frac{V}{4\pi }}\\, 1.99 inches

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