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# Solutions

## Solutions to Try Its

1. The graphs of $f\left(x\right)\\$ and $g\left(x\right)\\$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units. 2.
a)

b)
3. $g\left(x\right)=-f\left(x\right)\\$
 $x\\$ -2 0 2 4 $g\left(x\right)\\$ $-5\\$ $-10\\$ $-15\\$ $-20\\$

$h\left(x\right)=f\left(-x\right)\\$

 $x\\$ -2 0 2 4 $h\left(x\right)\\$ 15 10 5 unknown
4. even 5.
 $x\\$ 2 4 6 8 $g\left(x\right)\\$ 9 12 15 0
6. $g\left(x\right)=3x - 2\\$ 7. $g\left(x\right)=f\left(\frac{1}{3}x\right)\\$ so using the square root function we get $g\left(x\right)=\sqrt{\frac{1}{3}x}\\$ 8. 9. $g\left(x\right)=\frac{1}{x - 1}+1\\$ 10. Notice: $g\left(x\right)=f\left(-x\right)\\$ looks the same as $f\left(x\right)\\$ .

## Solution to Odd-Numbered Exercises

1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output. 3. A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output. 5. For a function $f$, substitute $\left(-x\right)\\$ for $\left(x\right)\\$ in $f\left(x\right)\\$. Simplify. If the resulting function is the same as the original function, $f\left(-x\right)=f\left(x\right)\\$, then the function is even. If the resulting function is the opposite of the original function, $f\left(-x\right)=-f\left(x\right)\\$, then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even. 7. $g\left(x\right)=|x - 1|-3\\$ 9. $g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2\\$ 11. The graph of $f\left(x+43\right)\\$ is a horizontal shift to the left 43 units of the graph of $f\\$. 13. The graph of $f\left(x - 4\right)\\$ is a horizontal shift to the right 4 units of the graph of $f\\$. 15. The graph of $f\left(x\right)+8\\$ is a vertical shift up 8 units of the graph of $f\\$. 17. The graph of $f\left(x\right)-7\\$ is a vertical shift down 7 units of the graph of $f\\$. 19. The graph of $f\left(x+4\right)-1\\$ is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of $f\\$. 21. decreasing on $\left(-\infty ,-3\right)\\$ and increasing on $\left(-3,\infty \right)\\$ 23. decreasing on $\left(0,\infty \right)\\$ 25. 27. 29. 31. $g\left(x\right)=f\left(x - 1\right),h\left(x\right)=f\left(x\right)+1\\$ 33. $f\left(x\right)=|x - 3|-2\\$ 35. $f\left(x\right)=\sqrt{x+3}-1\\$ 37. $f\left(x\right)={\left(x - 2\right)}^{2}\\$ 39. $f\left(x\right)=|x+3|-2\\$ 41. $f\left(x\right)=-\sqrt{x}\\$ 43. $f\left(x\right)=-{\left(x+1\right)}^{2}+2\\$ 45. $f\left(x\right)=\sqrt{-x}+1\\$ 47. even 49. odd 51. even 53. The graph of $g\\$ is a vertical reflection (across the $x\\$ -axis) of the graph of $f\\$. 55. The graph of $g\\$ is a vertical stretch by a factor of 4 of the graph of $f\\$. 57. The graph of $g\\$ is a horizontal compression by a factor of $\frac{1}{5}\\$ of the graph of $f\\$. 59. The graph of $g\\$ is a horizontal stretch by a factor of 3 of the graph of $f\\$. 61. The graph of $g\\$ is a horizontal reflection across the $y\\$ -axis and a vertical stretch by a factor of 3 of the graph of $f\\$. 63. $g\left(x\right)=|-4x|\\$ 65. $g\left(x\right)=\frac{1}{3{\left(x+2\right)}^{2}}-3\\$ 67. $g\left(x\right)=\frac{1}{2}{\left(x - 5\right)}^{2}+1\\$ 69. The graph of the function $f\left(x\right)={x}^{2}\\$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units. 71. The graph of $f\left(x\right)=|x|\\$ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up. 73. The graph of the function $f\left(x\right)={x}^{3}\\$ is compressed vertically by a factor of $\frac{1}{2}\\$. 75. The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units. 77. The graph of $f\left(x\right)=\sqrt{x}\\$ is shifted right 4 units and then reflected across the vertical line $x=4\\$. 79.
81.

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• Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..