Combinations and Compositions of Functions

1. [latex](f+g)(x)[/latex]: [latex]2x+6[/latex], domain: [latex](-\infty,\infty)[/latex]
   [latex](f-g)(x)[/latex]: [latex]2x^2+2x-6[/latex], domain: [latex](-\infty,\infty)[/latex]
   [latex](fg)(x)[/latex]: [latex]-x^4-2x^3+6x^2+12x[/latex], domain: [latex](-\infty,\infty)[/latex]
   [latex](\frac{f}{g})(x)[/latex]: [latex]\dfrac{x^2+2x}{6-x^2}[/latex], domain: [latex](-\infty,-\sqrt{6})\cup(-\sqrt{6},\sqrt{6})\cup(\sqrt{6},\infty)[/latex]
2. [latex](f+g)(x)[/latex]: [latex]\dfrac{4x^3+8x^2+1}{2x}[/latex], domain: [latex](-\infty,0)\cup(0,\infty)[/latex]
   [latex](f-g)(x)[/latex]: [latex]\dfrac{4x^3+8x^2-1}{2x}[/latex], domain: [latex](-\infty,0)\cup(0,\infty)[/latex]
   [latex](fg)(x)[/latex]: [latex]x+2[/latex], domain: [latex](-\infty,0)\cup(0,\infty)[/latex]
   [latex](\frac{f}{g})(x)[/latex]: [latex]4x^3+8x^2[/latex], domain: [latex](-\infty,0)\cup(0,\infty)[/latex]
3. [latex](f+g)(x)[/latex]: [latex]3x^2+\sqrt{x-5}[/latex], domain: [latex][5,\infty)[/latex]
   [latex](f-g)(x)[/latex]: [latex]3x^2-\sqrt{x-5}[/latex], domain: [latex][5,\infty)[/latex]
   [latex](fg)(x)[/latex]: [latex]3x^2\sqrt{x-5}[/latex], domain: [latex][5,\infty)[/latex]
   [latex](\frac{f}{g})(x)[/latex]: [latex]\dfrac{3x^2}{\sqrt{x-5}}[/latex], domain: [latex][5,\infty)[/latex]
   1. [latex]3[/latex]
   2. [latex]f(g(x))=2(3x-5)^2+1[/latex]
   3. [latex]f(g(x))=6x^2-2[/latex]
   4. [latex](g\circ g)(x)=3(3x-5)-5=9x-20[/latex]
   5. [latex](f\circ f)(-2)=163[/latex]
4. [latex]f(g(x))=\sqrt{x^2+3}+2[/latex],
   [latex]g(f(x))=x+4\sqrt{x}+7[/latex]
5. [latex]f(g(x))=\sqrt[3]{\dfrac{x+1}{x^3}}=\dfrac{\sqrt[3]{x+1}}{x}[/latex],
   [latex]g(f(x))=\dfrac{\sqrt[3]{x+1}}{x}[/latex]
6. [latex](f\circ g)(x)=\dfrac{1}{\frac{2}{x}+4-4}=\dfrac{x}{2}[/latex],
   [latex](g\circ f)(x)=2x-4[/latex]
7. [latex]f(g(h(x)))=(\dfrac{1}{x+3})^2+1[/latex]
   1. [latex](g\circ f)(x)=-\dfrac{3}{\sqrt{2-4x}}[/latex]
   2. [latex](-\infty,\dfrac{1}{2})[/latex]
   1. [latex](0,2)\cup(2,\infty)[/latex]
   2. [latex](-\infty,-2)\cup(2,\infty)[/latex]
   3. [latex](0,\infty)[/latex]
8. [latex](1,\infty)[/latex]
9. sample: [latex]f(x)=x^3[/latex] [latex]g(x)=x-5[/latex]
10. sample: [latex]f(x)=\dfrac{4}{x}[/latex] [latex]g(x)=(x+2)^2[/latex]
11. sample: [latex]f(x)=\sqrt[3]{x}[/latex] [latex]g(x)=\dfrac{1}{2x-3}[/latex]
12. sample: [latex]f(x)=\sqrt{x}[/latex] [latex]g(x)=\dfrac{3x-2}{x+5}[/latex]
13. sample: [latex]f(x)=\sqrt{x}[/latex] [latex]g(x)=2x+6[/latex]
14. sample: [latex]f(x)=\sqrt[3]{x}[/latex] [latex]g(x)=(x-1)[/latex]
15. sample: [latex]f(x)=x^3[/latex] [latex]g(x)=\dfrac{1}{x-2}[/latex]
16. sample: [latex]f(x)=\sqrt{x}[/latex] [latex]g(x)=\dfrac{2x-1}{3x+4}[/latex]
17. [latex]2[/latex]
18. [latex]5[/latex]
19. [latex]4[/latex]
20. [latex]0[/latex]
21. [latex]2[/latex]
22. [latex]1[/latex]
23. [latex]4[/latex]
24. [latex]4[/latex]
25. [latex]9[/latex]
26. [latex]4[/latex]
27. [latex]2[/latex]
28. [latex]3[/latex]
29. [latex]11[/latex]
30. [latex]0[/latex]
31. [latex]7[/latex]
32. [latex]f(g(0))=27[/latex], [latex]g(f(0))=-94[/latex]
33. [latex]f(g(0))=\dfrac{1}{5}[/latex], [latex]g(f(0))=5[/latex]
34. [latex]18x^2+60x+51[/latex]
35. [latex]g\circ g(x)=9x+20[/latex]
36. c
37. [latex]A(t)=\pi(25\sqrt{t}+2)^2[/latex] and [latex]A(2)=\pi(25\sqrt{4})^2=2500\pi[/latex] square inches
38. [latex]A(5)=\pi(2(5)+1)^2=121\pi[/latex] square units
    1. [latex]N(T(t))=23(5t+1.5)^2-56(5t+1.5)+1[/latex]
    2. [latex]3.38[/latex] hours

Transformations of Functions

1. [latex]g(x)=|x-1|-3[/latex]
2. [latex]g(x)=\dfrac{1}{(x+4)^2}+2[/latex]
3. The graph of [latex]f(x+43)[/latex] is a horizontal shift to the left [latex]43[/latex] units of the graph of [latex]f[/latex].
4. The graph of [latex]f(x-4)[/latex] is a horizontal shift to the right [latex]4[/latex] units of the graph of [latex]f[/latex].
5. The graph of [latex]f(x)+8[/latex] is a vertical shift up [latex]8[/latex] units of the graph of [latex]f[/latex].
6. The graph of [latex]f(x)-7[/latex] is a vertical shift down [latex]7[/latex] units of the graph of [latex]f[/latex].
7. The graph of [latex]f(x+4)-1[/latex] is a horizontal shift to the left [latex]4[/latex] units and a vertical shift down [latex]1[/latex] unit of the graph of [latex]f[/latex].
8. decreasing on latex](-\infty,-3)[/latex] and increasing on latex](-3,\infty)[/latex]
9. decreasing on latex](0,\infty)[/latex]
10. 
11. 
12. 
13. [latex]g(x)=f(x-1)[/latex], [latex]h(x)=f(x)+1[/latex]
14. [latex]f(x)=|x-3|-2[/latex]
15. [latex]f(x)=\sqrt{x+3}-1[/latex]
16. [latex]f(x)=(x-2)^2[/latex]
17. [latex]f(x)=|x+3|-2[/latex]
18. [latex]f(x)=-\sqrt{x}[/latex]
19. [latex]f(x)=-(x+1)^2+2[/latex]
20. [latex]f(x)=\sqrt{-x}+1[/latex]
21. even
22. odd
23. even
24. The graph of [latex]g[/latex] is a vertical reflection (across the x-axis) of the graph of [latex]f[/latex].
25. The graph of [latex]g[/latex] is a vertical stretch by a factor of [latex]4[/latex] of the graph of [latex]f[/latex].
26. The graph of [latex]g[/latex] is a horizontal compression by a factor of [latex]\dfrac{1}{5}[/latex] of the graph of [latex]f[/latex].
27. The graph of [latex]g[/latex] is a horizontal stretch by a factor of [latex]3[/latex] of the graph of [latex]f[/latex].
28. The graph of [latex]g[/latex] is a horizontal reflection across the [latex]y[/latex]-axis and a vertical stretch by a factor of [latex]3[/latex] of the graph of [latex]f[/latex].
29. [latex]g(x)=|-4x|[/latex]
30. [latex]g(x)=-\dfrac{1}{3(x+2)^2}-3[/latex]
31. [latex]g(x)=\dfrac{1}{2}(x-5)^2+1[/latex]
32. The graph of the function [latex]f(x)=x^2[/latex] is shifted to the left [latex]1[/latex] unit, stretched vertically by a factor of [latex]4[/latex], and shifted down [latex]5[/latex] units.
    
33. The graph of [latex]f(x)=|x|[/latex] is stretched vertically by a factor of [latex]2[/latex], shifted horizontally [latex]4[/latex] units to the right, reflected across the horizontal axis, and then shifted vertically [latex]3[/latex] units up.
    
34. The graph of the function [latex]f(x)=x^3[/latex] is compressed vertically by a factor of [latex]\dfrac{1}{2}[/latex].
    
35. The graph of the function is stretched horizontally by a factor of [latex]3[/latex] and then shifted vertically downward by [latex]3[/latex] units.
    
36. The graph of [latex]f(x)=\sqrt{x}[/latex] is shifted right [latex]4[/latex] units and then reflected across the vertical line [latex]x=4[/latex].
    
37. 
38. 

Inverse Functions

1. [latex]f^{-1}(x)=x-3[/latex]
2. [latex]f^{-1}(x)=2-x[/latex]
3. [latex]f^{-1}(x)=\dfrac{-2x}{x-1}[/latex]
4. domain of [latex]f(x):[−7,\infty)[/latex]; [latex]f^{-1}(x)=\sqrt{x}-7[/latex]
5. domain of [latex]f(x):[0,\infty)[/latex]; [latex]f^{-1}(x)=\sqrt{x+5}[/latex]
6. [latex]f(g(x))=x[/latex], [latex]g(f(x))=x[/latex]
7. one-to-one
8. one-to-one
9. not one-to-one
10. [latex]3[/latex]
11. [latex]2[/latex]
12. 
13. [latex][2,10][/latex]
14. [latex]6[/latex]
15. [latex]-4[/latex]
16. [latex]0[/latex]
17. [latex]1[/latex]

18. [latex]x[/latex]	[latex]1[/latex]	[latex]4[/latex]	[latex]7[/latex]	[latex]12[/latex]	[latex]16[/latex]
    [latex]f^{-1}(x)[/latex]	[latex]3[/latex]	[latex]6[/latex]	[latex]9[/latex]	[latex]13[/latex]	[latex]14[/latex]

19. [latex]f^{-1}(x)=\dfrac{5}{9}(x-32)[/latex]. Given the Fahrenheit temperature, [latex]x[/latex], this formula allows you to calculate the Celsius temperature.
20. [latex]t(d)=\dfrac{d}{50}[/latex], [latex]t(180)=\dfrac{180}{50}[/latex]. The time for the car to travel [latex]180[/latex] miles is [latex]3.6[/latex] hours.