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Study Guides > MATH 1314: College Algebra

Solutions

Solutions for Try Its

1. {(fg)(x)=f(x)g(x)=(x1)(x21)=x3x2x+1(fg)(x)=f(x)g(x)=(x1)(x21)=xx2\begin{cases}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)=\left(x - 1\right)\left({x}^{2}-1\right)={x}^{3}-{x}^{2}-x+1\\ \left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)=\left(x - 1\right)-\left({x}^{2}-1\right)=x-{x}^{2}\end{cases} No, the functions are not the same. 2. A gravitational force is still a force, so a(G(r))a\left(G\left(r\right)\right) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F))G\left(a\left(F\right)\right) does not make sense. 3. f(g(1))=f(3)=3f\left(g\left(1\right)\right)=f\left(3\right)=3 and g(f(4))=g(1)=3g\left(f\left(4\right)\right)=g\left(1\right)=3 4. g(f(2))=g(5)=3g\left(f\left(2\right)\right)=g\left(5\right)=3 5. A. 8; B. 20 6. [4,0)(0,)\left[-4,0\right)\cup \left(0,\infty \right) 7. Possible answer:

g(x)=4+x2g\left(x\right)=\sqrt{4+{x}^{2}}

h(x)=43xh\left(x\right)=\frac{4}{3-x}

f=hgf=h\circ g

   

Solutions to Odd-Numbered Exercises

1. Find the numbers that make the function in the denominator gg equal to zero, and check for any other domain restrictions on ff and gg, such as an even-indexed root or zeros in the denominator. 3. Yes. Sample answer: Let f(x)=x+1 and g(x)=x1f\left(x\right)=x+1\text{ and }g\left(x\right)=x - 1. Then f(g(x))=f(x1)=(x1)+1=xf\left(g\left(x\right)\right)=f\left(x - 1\right)=\left(x - 1\right)+1=x and g(f(x))=g(x+1)=(x+1)1=xg\left(f\left(x\right)\right)=g\left(x+1\right)=\left(x+1\right)-1=x. So fg=gff\circ g=g\circ f. 5. (f+g)(x)=2x+6\left(f+g\right)\left(x\right)=2x+6, domain: (,)\left(-\infty ,\infty \right) (fg)(x)=2x2+2x6[/latex],domain:[latex](,)\left(f-g\right)\left(x\right)=2{x}^{2}+2x - 6[/latex], domain: [latex]\left(-\infty ,\infty \right) (fg)(x)=x42x3+6x2+12x[/latex],domain:[latex](,)\left(fg\right)\left(x\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x[/latex], domain: [latex]\left(-\infty ,\infty \right) (fg)(x)=x2+2x6x2[/latex],domain:[latex](,6)(6,6)(6,)\left(\frac{f}{g}\right)\left(x\right)=\frac{{x}^{2}+2x}{6-{x}^{2}}[/latex], domain: [latex]\left(-\infty ,-\sqrt{6}\right)\cup \left(-\sqrt{6},\sqrt{6}\right)\cup \left(\sqrt{6},\infty \right) 7. (f+g)(x)=4x3+8x2+12x\left(f+g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}+1}{2x}, domain: (,0)(0,)\left(-\infty ,0\right)\cup \left(0,\infty \right) (fg)(x)=4x3+8x212x[/latex],domain:[latex](,0)(0,)\left(f-g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}-1}{2x}[/latex], domain: [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right) (fg)(x)=x+2[/latex],domain:[latex](,0)(0,)\left(fg\right)\left(x\right)=x+2[/latex], domain: [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right) (fg)(x)=4x3+8x2[/latex],domain:[latex](,0)(0,)\left(\frac{f}{g}\right)\left(x\right)=4{x}^{3}+8{x}^{2}[/latex], domain: [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right) 9. (f+g)(x)=3x2+x5\left(f+g\right)\left(x\right)=3{x}^{2}+\sqrt{x - 5}, domain: [5,)\left[5,\infty \right) (fg)(x)=3x2x5[/latex],domain:[latex][5,)\left(f-g\right)\left(x\right)=3{x}^{2}-\sqrt{x - 5}[/latex], domain: [latex]\left[5,\infty \right) (fg)(x)=3x2x5[/latex],domain:[latex][5,)\left(fg\right)\left(x\right)=3{x}^{2}\sqrt{x - 5}[/latex], domain: [latex]\left[5,\infty \right) (fg)(x)=3x2x5[/latex],domain:[latex](5,)\left(\frac{f}{g}\right)\left(x\right)=\frac{3{x}^{2}}{\sqrt{x - 5}}[/latex], domain: [latex]\left(5,\infty \right) 11. a. 3; b. f(g(x))=2(3x5)2+1f\left(g\left(x\right)\right)=2{\left(3x - 5\right)}^{2}+1; c. f(g(x))=6x22f\left(g\left(x\right)\right)=6{x}^{2}-2; d. (gg)(x)=3(3x5)5=9x20\left(g\circ g\right)\left(x\right)=3\left(3x - 5\right)-5=9x - 20; e. (ff)(2)=163\left(f\circ f\right)\left(-2\right)=163 13. f(g(x))=x2+3+2,g(f(x))=x+4x+7f\left(g\left(x\right)\right)=\sqrt{{x}^{2}+3}+2,g\left(f\left(x\right)\right)=x+4\sqrt{x}+7 15. f(g(x))=x+1x33=x+13x,g(f(x))=x3+1xf\left(g\left(x\right)\right)=\sqrt[3]{\frac{x+1}{{x}^{3}}}=\frac{\sqrt[3]{x+1}}{x},g\left(f\left(x\right)\right)=\frac{\sqrt[3]{x}+1}{x} 17. (fg)(x)=12x+44=x2, (gf)(x)=2x4\left(f\circ g\right)\left(x\right)=\frac{1}{\frac{2}{x}+4 - 4}=\frac{x}{2},\text{ }\left(g\circ f\right)\left(x\right)=2x - 4 19. f(g(h(x)))=(1x+3)2+1f\left(g\left(h\left(x\right)\right)\right)={\left(\frac{1}{x+3}\right)}^{2}+1 21. a. (gf)(x)=324x\left(g\circ f\right)\left(x\right)=-\frac{3}{\sqrt{2 - 4x}}; b. (,12)\left(-\infty ,\frac{1}{2}\right) 23. a. (0,2)(2,)\left(0,2\right)\cup \left(2,\infty \right); b. (,2)(2,)\left(-\infty ,-2\right)\cup \left(2,\infty \right); c. (0,)\left(0,\infty \right) 25. (1,)\left(1,\infty \right) 27. sample: {f(x)=x3g(x)=x5\begin{cases}f\left(x\right)={x}^{3}\\ g\left(x\right)=x - 5\end{cases} 29. sample: {f(x)=4xg(x)=(x+2)2\begin{cases}f\left(x\right)=\frac{4}{x}\hfill \\ g\left(x\right)={\left(x+2\right)}^{2}\hfill \end{cases} 31. sample: {f(x)=x3g(x)=12x3\begin{cases}f\left(x\right)=\sqrt[3]{x}\\ g\left(x\right)=\frac{1}{2x - 3}\end{cases} 33. sample: {f(x)=x4g(x)=3x2x+5\begin{cases}f\left(x\right)=\sqrt[4]{x}\\ g\left(x\right)=\frac{3x - 2}{x+5}\end{cases} 35. sample: f(x)=xf\left(x\right)=\sqrt{x} g(x)=2x+6g\left(x\right)=2x+6 37.sample: f(x)=x3f\left(x\right)=\sqrt[3]{x} g(x)=(x1)g\left(x\right)=\left(x - 1\right) 39. sample: f(x)=x3f\left(x\right)={x}^{3} g(x)=1x2g\left(x\right)=\frac{1}{x - 2} 41. sample: f(x)=xf\left(x\right)=\sqrt{x} g(x)=2x13x+4g\left(x\right)=\frac{2x - 1}{3x+4} 43. 2 45. 5 47. 4 49. 0 51. 2 53. 1 55. 4 57. 4 59. 9 61. 4 63. 2 65. 3 67. 11 69. 0 71. 7 73. f(g(0))=27,g(f(0))=94f\left(g\left(0\right)\right)=27,g\left(f\left(0\right)\right)=-94 75. f(g(0))=15,g(f(0))=5f\left(g\left(0\right)\right)=\frac{1}{5},g\left(f\left(0\right)\right)=5 77. 18x2+60x+5118{x}^{2}+60x+51 79. gg(x)=9x+20g\circ g\left(x\right)=9x+20 81. 2 83. (,)\left(-\infty ,\infty \right) 85. False 87. (fg)(6)=6\left(f\circ g\right)\left(6\right)=6 ; (gf)(6)=6\left(g\circ f\right)\left(6\right)=6 89. (fg)(11)=11,(gf)(11)=11\left(f\circ g\right)\left(11\right)=11,\left(g\circ f\right)\left(11\right)=11 91. c. Solve A(m(t))=4A\left(m\left(t\right)\right)=4. 93. A(t)=π(25t+2)2A\left(t\right)=\pi {\left(25\sqrt{t+2}\right)}^{2} and A(2)=π(254)2=2500πA\left(2\right)=\pi {\left(25\sqrt{4}\right)}^{2}=2500\pi square inches 95. A(5)=π(2(5)+1)2=121πA\left(5\right)=\pi {\left(2\left(5\right)+1\right)}^{2}=121\pi square units 97. a. N(T(t))=23(5t+1.5)256(5t+1.5)+1N\left(T\left(t\right)\right)=23{\left(5t+1.5\right)}^{2}-56\left(5t+1.5\right)+1; b. 3.38 hours

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