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4.2e: Exercises - Graphs of Exponential Functions

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    A: Concepts

    Example \(\PageIndex{A}\) 

    1) What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

    2) What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

    3) Explore and discuss the graphs of  \(F(x)=(b)^x\) and \(G(x)=\left ( \frac{1}{b} \right )^x\). Then make a conjecture about the relationship between the graphs of the functions \(b^x\) and \(\left ( \frac{1}{b} \right )^x\) for any real number \(b>0\).

    4) Prove the conjecture made in the previous exercise.

    5) Explore and discuss the graphs of \(f(x) = 4^x\), \(g(x)=4^{x-2}\), and \(h(x)=\left ( \frac{1}{16} \right )4^x\). Then make a conjecture about the relationship between the graphs of the functions \(b^x\) and \(\left ( \frac{1}{b^n} \right )b^x\) for any real number \(n\) and real number \(b>0\).

    6) Prove the conjecture made in the previous exercise.

    For the following exercises, match each function with one of the graphs in Figure at the right.

     

    7) \(f(x)=2(0.69)^x \\[3pt] \)

    8) \(f(x)=2(1.28)^x \\[3pt] \)

    9) \(f(x)=2(0.81)^x \\[3pt] \)

    10) \(f(x)=4(1.28)^x \\[3pt] \)

    11) \(f(x)=2(1.59)^x \\[3pt] \)

    12) \(f(x)=4(0.69)^x\)

    		 CNX_PreCalc_Figure_04_02_206.jpg

    For the following exercises, use the graphs shown in Figure at the right. All have the form \(f(x)=ab^x\).

     

    13) Which graph has the largest value for \(b? \\[3pt] \)

    14) Which graph has the smallest value for \(b? \\[3pt] \)

    15) Which graph has the largest value for \(a? \\[3pt] \)

    16) Which graph has the smallest value for \(a\)?\

    		 CNX_PreCalc_Figure_04_02_207.jpg
    Answers to odd exercises:

    1. An asymptote is a line that the graph of a function approaches, as \(x\) either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

    3. The graph of \(G(x)=\left ( \frac{1}{b} \right )^x\) is the refelction about the \(y\)-axis of the graph of \(F(x)=(b)^x\); For any real number \(b>0\) and function \(f(x)=(b)^x\) , the graph of \(\left ( \frac{1}{b} \right )^x\) is the the reflection about the \(y\)-axis, \(F(-x)\).

    5. \(g(x)\) and \(h(x)\) are the same graph and are a horizontal shift to the right of the graph of \(f(x)\); For any real number \(n\), real number \(b>0\), and function \(f(x)=b^x\), the graph of \(\left ( \frac{1}{b^n} \right )b^x\) is the horizontal shift \(f(x-n)\).

    7. B, 9. A, 11. E, 13. D, 15. C

    B: Match Graphs With Equations

    Exercise \(\PageIndex{B}\) 

    \( \bigstar \) In the following exercises, match the graphs to one of the following functions:

    a. \(2^{x}\) b. \(2^{x+1}\) c. \(2^{x-1}\) d. \(2^{x}+2\) e. \(2^{x}-2\) f. \(3^{x}\)

    17.

    This figure shows an exponential that passes through (1, 1 over 3), (0, 1), and (1, 3). 

    18.

    This figure shows an exponential that passes through (negative 2, 1 over 2), (negative 1, 1), and (0, 2).

    19.

    This figure shows an exponential that passes through (1, 1 over 2), (0, 1), and (1, 2).

    20.

    This figure shows an exponential that passes through (0, 1 over 2), (1, 1), and (2, 2). 

    21.

    This figure shows an exponential that passes through (negative 1, 3 over 2), (0, negative 1), and (1, 0). 

    22.

    This figure shows an exponential that passes through (negative 1, 5 over 2), (0, 3), and (1, 4).

    \( \bigstar \) For the following exercises, match the exponential equation to the correct graph.

    a. \(y=4^{−x}\) b. \(y=3^{x−1}\) c. \(y=2^{x+1}\) d. \(y=(\frac{1}{2})^x+2\) e. \(y=−3^{−x}\) f. \(y=1−5^x\)

    23)

    4.2E #23.png

    24)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, 1). Another point of the graph is at (-1, 4).

    25)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, (1/3)). Another point of the graph is at (1, 1).

    26)

    4.2E #26.png

    27)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, -1). Another point of the graph is at (-1, -3).

    28)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 2). Another point of the graph is at (-1, 1).
    Answers to odd exercises:

    17. f,   19. a,   21. e;   23. d,   25. b,   27. e

    C: Graph Basic Exponential Functions

    Exercise \(\PageIndex{C}\) 

    \( \bigstar \) In the following exercises, graph each exponential function.

    29. \(f(x)=2^{x}\)

    30. \(g(x)=3^{x}\)

    31. \(f(x)=6^{x}\)

    32. \(g(x)=7^{x}\)

    33. \(f(x)=(1.5)^{x}\)

    34. \(g(x)=(2.5)^{x}\)

    35. \(f(x)=\left(\frac{1}{2}\right)^{x}\)

    36. \(g(x)=\left(\frac{1}{3}\right)^{x}\)

    37. \(f(x)=\left(\frac{1}{6}\right)^{x}\)

    38. \(g(x)=\left(\frac{1}{7}\right)^{x}\)

    39. \(f(x)=(0.4)^{x}\)

    40. \(g(x)=(0.6)^{x}\)

    41. \(f(x)=4^{x}\)

    42. \(g(x)=5^{x}\)

    43. \(f(x)=\left(\frac{1}{4}\right)^{x}\)

    44. \(h(x)=\left(\frac{1}{5}\right)^{x}\)
    Answers to odd exercises:

    29.

    This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).
    \(f(x)=2^{x}\)

    31.

    This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).
    \(f(x)=6^{x}\)

    33.

    This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).
    \(f(x)=(1.5)^{x}\)

    35.

    This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).
    \(f(x)=\left(\frac{1}{2}\right)^{x}\)

    37.

    This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).
    \(f(x)=\left(\frac{1}{6}\right)^{x}\)

    39.

    This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).
    \(f(x)=(0.4)^{x}\)

    41. 

    35fd614da9fe5909d018e87da5e5ca07.png
    \(f(x)=4^{x}\)

    43. 

    d7f05216b6cd46049c56f0b9b2ed16fe.png
    \(f(x)=\left(\frac{1}{4}\right)^{x}\)
    		  

    D: Graph Shifts of Exponential Functions

    Exercise \(\PageIndex{D}\) 

    \( \bigstar \) In the following exercises, use transformations to graph each exponential function. State the transformations that must be done to the parent function in order to obtain the graph.

    45. \( g(x)=2^{x}+1\)

    46.\( g(x)=2^{x}-1\)

    47. \( g(x)=2^{x-2}\)

    48. \( g(x)=2^{x+2}\) 

    49. \(g(x)=3^{x}+2\)

    50. \(f(x)=3^{x}-6\)

    51. \( g(x)=4^{x-1}\)

    52. \(f(x)=4^{x+2}\)

    53. \(f(x)=2^{x}+3\)

    54. \(f(x)=2^{x}-3\)

    55. \(f(x)=3^{x+2}\)

    56. \(f(x)=3^{x+1}\) 

    57. \(h(x)=2^{x-3}-2\)

    58. \(g(x)=3^{x-1}+3\) 

    59. \(f(x)=3^{x+1}-4\)

    60. \(h(x)=3^{x-2}+4\)

    61. \(f(x)=4^{x}+2\)

    62. \(f(x)=4^x−1\)

    63. \(f(x)=5^{x+1}+2\)

    64. \(f(x)=10^{x-4}+2\)
    Answers to odd exercises:

    45. Up 1

    This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1), and (1, 2). The second function g of x equals 2 to the x power plus 1 is marked in red and passes through the points (negative 1, 1), (0, 2), and (1, 4).\(g(x)=2^{x}+1\)

    47. Right 2

    This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).
    \( g(x)=2^{x-2}\)

    49. Up 2

    This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).
    \(g(x)=3^{x}+2\)

    51. Right 1

    This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).
    \(g(x)=4^{x-1}\)

    53. Up 3

    This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).
    \(f(x)=2^{x}+3\)

    55. Left 2

    This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).
    \(f(x)=3^{x+2}\)

    57. Right 3, Down 2;
    Domain: \((-\infty, \infty)\); Range: \((-2, \infty)\)

    e65b0fd52f89b20c5be372ff5bd39859.png
    \(h(x)=2^{x-3}-2\)

    59. Left 1, Down 4;
    Domain: \((-\infty, \infty)\); Range: \((-4, \infty)\)

    3dc90132f8492aa3db94575b0e52d3b8.png
    \(f(x)=3^{x+1}-4\)

    61. up 2; Domain: \((-\infty, \infty)\); Range: \((2, \infty)\)

    3f39f313029712673a894c3e3d3cabfd.png
    \(f(x)=4^{x}+2\)

    63. Left 1, up 2

    CNX_Calc_Figure_01_05_212.jpeg
    63. \(f(x)=5^{x+1}+2\)
    		    

    \( \bigstar \) In the following exercises, use transformations to graph each exponential function. State the transformations that must be done to the parent function in order to obtain the graph.

    65. \(f(x)=\left(\frac{1}{2}\right)^{x-4}\)

    66. \(f(x)=\left(\frac{1}{2}\right)^{x}-3\)

    67. \(f(x)=\left(\frac{1}{4}\right)^{x}-2\)

    68. \(h(x)=\left(\frac{1}{3}\right)^{x}+2\)

    69. \(f(x)=e^{x}-3\)

    70. \(f(x)=e^{x}+2\)

    71. \(f(x)=e^{x}+1\)

    72. \(f(x)=e^{x-2}\)

    73. \(f(x)=e^{x+1}\)

    74. \(f(x)=e^{x-3}\)

     

    75. \(f(x)=e^{x-2}+1\)

    76. \(f(x)=e^{x+2}-1\)
    Answers to odd exercises:

    65. Right 4

    This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).
    \(f(x)=\left(\frac{1}{2}\right)^{x-4}\)

    67.  Down 2; Domain: \((-\infty, \infty)\); Range: \((-2, \infty)\)

    d635b8f97a4b404d71b4030889d9c2fd.png
    \(f(x)=\left(\frac{1}{4}\right)^{x}-2\)

    69. Down 3. Domain: \((-\infty, \infty)\); Range: \((-3, \infty)\)

    d38859f7a31bd1d0488e2677157f879d.png
    \(f(x)=e^{x}-3\)

    71. Up 1

    This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).
    \(f(x)=e^{x}+1\)

    73. Left 1; Domain: \((-\infty, \infty)\); Range: \((0, \infty)\)

    ee11942a039611258f662587981d688e.png
    \(f(x)=e^{x+1}\)

    75. Right 2, up 1; Domain: \((-\infty, \infty)\); Range: \((1, \infty)\)

    3ceac41515b5a247cd6f0e04bbcf4565.png
    \(f(x)=e^{x-2}+1\)

    E: Graph Reflections and Stretches of Exponential Functions

    Exercise \(\PageIndex{E}\) 

    \( \bigstar \) In the following exercises, graph each exponential function. State the transformations that must be done to the parent function in order to obtain the graph and state its domain, range, and horizontal asymptote.

    77. \(f(x)=-2^{x}\)

    78. \(f(x)=2^{-x-1}-1\)

    79. \(g(x)=2^{-x}-3\)

    80. \(g(x)=3^{-x}+1\)

    81. \(f(x)=6-10^{-x}\)

    82. \(g(x)=5-4^{-x}\)

    83. \(f(x)=5-2^{x}\)

    84. \(f(x)=3-3^{x}\)

    84.1 \(f(x) = 3 \cdot 2^{x-4}-5 \)

    84.2 \(y = 5-\frac{1}{2} \cdot 4^{x-3} \)

    84.3 \(f(x) = 3^{x/2+3}-4\)

    84.4 \(f(x) = 5-2^{-x/3+1}\)

    85) \(f(x)=1−2^{−x}\)

    86) \(f(x)=2^{-x}\)

    87. \(g(x)=-e^{x}\)

    88. \(g(x)=e^{-x}\)

    89. \(h(x)=-e^{x+1}\)

    90. \(h(x)=-e^{x}+3\)

    91) \(f(x)=e^{−x}−1\)

    \( \bigstar \) For the following exercises, graph each set of functions on the same axes.

    93) \(f(x)=3\left ( \frac{1}{4} \right )^x, g(x)=3(2)^x, h(x)=3(4)^x\) 94) \(f(x)=\frac{1}{4}(3)^x, g(x)=2(3)^x, h(x)=4(3)^x\)

    \( \bigstar \) For the following exercises, describe the end behavior of the graphs of the functions and the transformations needed to do on the parent graph to obtain the graph of the function.

    95) \(f(x)=-5(4)^x-1\) 96) \(f(x)=3\left ( \frac{1}{2} \right )^x-2\) 97) \(f(x)=3(4)^{-x}+2\)
    Answers to odd exercises:

    77. Reflect over x-axis
    HA: y = 0
    Domain: \((-\infty, \infty)\); Range: \((-\infty, 0)\)

    This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).
    \(f(x)=-2^{x}\)

    79. Reflect over y-axis, down 3;
    HA: y = -3
    Domain: \((-\infty, \infty)\); Range: \((-3, \infty)\)

    77a760d793646fddf6195692ee0c0eae.png
    \(g(x)=2^{-x}-3\)

    81. Reflect over y-axis, reflect over x-axis, up 6; HA: y = 6; Domain: \((-\infty, \infty)\); Range: \((-\infty, 6)\)

    e09e4075efad1f038e5b8cb008abb768.png
    \(f(x)=6-10^{-x}\)

    83. Reflect over x-axis, up 5; HA: y = 5; Domain: \((-\infty, \infty)\); Range: \((-\infty, 5)\)

    6f7e6b277d136faa0411780695ecb29d.png
    \(f(x)=5-2^{x}\)

    85. Reflect over y-axis, repfect over x-axis up 1; HA: y=1; Domain: all real numbers, range: \((−∞,1) \)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the line “y = 1” without touching it. There x intercept and the y intercept are both at the origin. Another point of the graph is at (-1, -1).\(f(x)=1−2^{−x}\)

    87. Reflect over x-axis; HA: y = 0; Domain: \((-\infty, \infty)\); Range: \((-\infty, 0)\)

    f5f920ae438f0ffa3668a3ae55926631.png
    \(g(x)=-e^{x}\)

    89. Left 1, Reflect over x-axis; HA: y=0; Domain: \((-\infty, \infty)\); Range: \((-\infty, 0)\)

    ff540e17defeac3b805f703e40e07252.png
    \(h(x)=-e^{x+1}\)

    91. Reflect over y-axis, down 1; HA: y = -1; Domain: all real numbers, range: \((−1,∞)\)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the line “y = -1” without touching it. There x intercept and the y intercept are both at the origin. There is an approximate point on the graph at (-1, 1.7).
    \(f(x)=e^{−x}−1\)

    93.

    CNX_PreCalc_Figure_04_02_204.jpg 

    95. As \(x\rightarrow \infty , f(x)\rightarrow -\infty\); As \(x\rightarrow -\infty , f(x)\rightarrow -1\); Reflect over x-axis, y's stretched by a factor of 5, down 1 
    97. As \(x\rightarrow \infty , f(x)\rightarrow 2\); As \(x\rightarrow -\infty , f(x)\rightarrow \infty\); Reflect over y-axis, y's get stretched by a factor of 3, up 2.

    F: Construct an Equation Given a Description

    Exercise \(\PageIndex{F}\) 

    \( \bigstar \) For the following exercises, start with the graph of \(f(x)=4^x\). Then write a function that results from the given transformation.

    98) Shift \(f(x)\) \(4\) units upward

    99) Shift \(f(x)\) \(3\) units downward

    100) Shift \(f(x)\) \(2\) units left

    101) Shift \(f(x)\) \(5\) units right

    102) Reflect \(f(x)\) about the \(x\)-axis

    103) Reflect \(f(x)\) about the \(y\)-axis

    \( \bigstar \) For the following exercises, (a) graph the function, (b) graph its reflection about the \(y\)-axis and construct the function for the reflected graph, and (c) give the \(y\)-intercept.

    104) \(f(x)=3\left ( \frac{1}{2} \right )^x\) 105) \(g(x)=-2(0.25)^x\) 106) \(h(x)=6(1.75)^{-x}\)

    \( \bigstar \) For the following exercises, (a) graph the function, (b) graph its reflection about the \(x\)-axis and construct the function for the reflected graph.

    107) \(f(x)=\frac{1}{2}(4)^x\) 108) \(f(x)=3(0.75)^x-1\) 109) \(f(x)=-4(2)^x+2\)

    \( \bigstar \) For the following exercises, (a) construct the equation of the exponential function given a description. (b) State its \(y\)-intercept, domain, and range.

    111) The graph of \(f(x)=10^x\) is reflected about the \(x\)-axis and shifted upward \(7\) units. What is the equation of the new function, \(g(x)\)? State its \(y\)-intercept, domain, and range.

    112) The graph of \(f(x)=\left ( \frac{1}{2} \right )^{-x}\) is reflected about the \(y\)-axis and compressed vertically by a factor of \(\left ( \frac{1}{5} \right )\) What is the equation of the new function, \(g(x)\) State its \(y\)-intercept, domain, and range.

    113) The graph of \(f(x) = 3^x\) is reflected about the \(y\)-axis and stretched vertically by a factor of \(4\). What is the equation of the new function, \(g(x)\). State its \(y\)-intercept, domain, and range.

    114) The graph of \(f(x)=(1.68)^x\) is shifted right \(3\) units, stretched vertically by a factor of \(2\), reflected about the \(x\)-axis, and then shifted downward \(3\) units. What is the equation of the new function, \(g(x)\)? State its \(y\)-intercept (to the nearest thousandth), domain, and range.

    115) The graph of \(f(x)=2\left ( \frac{1}{4} \right )^{x-1}\) is shifted left \(2\) units, stretched vertically by a factor of \(4\), reflected about the \(x\)-axis, and then shifted downward \(4\) units. What is the equation of the new function, \(g(x)\)? State the y-intercept, domain, and range of \(g(x)\).

    Answers to odd exercises:

    99. \(f(x)=4^x-3\)  \(\qquad\) 101. \(f(x)=4^{x-5}\)  \(\qquad\) 103. \(f(x)=4^{-x}\)

    105.

    CNX_PreCalc_Figure_04_02_202.jpg

    \(y\)-intercept: \((0,-2)\)

    107

    CNX_PreCalc_Figure_04_02_208.jpg

    109

    CNX_PreCalc_Figure_04_02_210.jpg

    111. \(g(x)=-10^x + 7\); \(y\)-intercept: \((0,6)\);  Domain: all real numbers; Range: all real numbers less than \(7\).

    113. \(g(x)=4(3)^{-x}\) \((0,4)\) Domain: all real numbers; Range: all real numbers greater than \(0\).

    115. \(g(x)=-8 \left ( \frac{1}{4} \right )^{x+1} -4\); \(y\)-intercept: \((0,-6)\);   Domain: \((-\infty, \infty)\); Range: \((-\infty, -4)\)

    G:  Construct an Equation Given a Graph

    Exercise \(\PageIndex{G}\) 

    \( \bigstar \) Write an equation for each of the functions graphed below. The graphs #117-123 are transformations of the parent function \(f(x)=2^x\). The parent functions for #124-127 must be determined.

    117.

    CNX_PreCalc_Figure_04_02_215.jpg

    118.

    CNX_Calc_Figure_01_05_208.jpeg

    119.

    CNX_PreCalc_Figure_04_02_212.jpg

    120.

    b584465c1351726169187e6a796b2e59.png

    121.

    CNX_PreCalc_Figure_04_02_216.jpg

    122.

    CNX_PreCalc_Figure_04_02_214.jpg

    123.

    CNX_PreCalc_Figure_04_02_218.jpg

    124.

    4.2e #124.png

    125.

    CNX_Calc_Figure_01_05_210.jpeg

    126.

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3). Another point of the graph is at (-1, 1).

    127. (Hint: this graph has been stretched)

    CNX_PreCalc_Figure_04_02_217.jpg

     
    Answer to odd exercises:

    117. \(y=-2^x+3\)

    119. \(y=2^x+3\)

    121. \(y=-2^{-x+1}+3\)

    123. \(y=2^{-x+1}-4\)

    125. \(y=4^x-1\)

    127. \(y=-2(3)^x+7\)

    H:  Practice with Exponent Properties

    Exercise \(\PageIndex{H}\) 

    \( \bigstar \) Write the following with only one exponent, \(x\). Simplify.

    1. \(2^{x+3} \)
    2. \(7^{2+x} \)
    3. \( 4^{x-3} \)
    4. \( 9^{x-2} \)
    5. \( 4^{4-x} \)
    6. \( 6^{2-x} \)
    1. \(5^{2x} \)
    2. \(4^ {2x} \)
    3. \( 5^x \cdot 7^x \)
    4. \( 9^x \cdot 4^x \)
    5. \( 2^x \cdot 3^{2x} \)
    6. \( x^2 \cdot y^2 \)
    1. \( 2^{3x+2} \)
    2. \( 3^{2x+3} \)
    3. \( 4^{3x-2} \)
    4. \( 2^{5x-3} \)
    5. \( 6^{x+1} \cdot 4^{x-2} \)
    6. \(8^{2x+1} \cdot 7^{x-1} \)
    1. \( \dfrac{ 3^x}{5^x} \\[3pt] \)
    2. \( \dfrac{ 4^x}{w^x} \\[3pt] \)
    3. \( \dfrac{ 2^{4x}}{8^x} \)
    1. \( \dfrac{ 9^x}{6^{2x}} \\[3pt] \)
    2. \( \dfrac{ 3^{x+1}}{2^{ 2x-3 }} \\[3pt] \)
    3. \( \dfrac{ 8^{x-1}}{4^{3x+2}} \)
    1. \( \dfrac{ 6^{x+2}}{4^{2x+3}} \\[3pt] \)
    2. \( \dfrac{ 5^{2x-1}}{3^{3x-2}} \)
    Answers to odd exercises:

    131.  \( 8(2^x)  \quad \) 133.  \( \dfrac{4^x}{64}  \quad \) 135.  \( \dfrac{256}{4^x} = 256 \left( \dfrac{1}{4} \right)^x  \quad \) 137.  \( 25^x  \quad \) 139.  \(35^x \) \(\quad\) 141.  \( 18^x  \quad \) 143.  \(4(8^x) \)
    145.  \( \dfrac{64^x}{16}  \quad \) 147.  \(  \dfrac{3}{8}24^x  \quad \) 149.  \(\left( \dfrac{3}{5} \right)^x\)  \(\quad\) 151.  \( 2^x  \quad \) 153.  \(24 \left( \dfrac{3}{4} \right)^x  \quad \) 155.   \(\dfrac{9}{16} \left( \dfrac{3}{8} \right)^x \)

    4.2e: Exercises - Graphs of Exponential Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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