AP PreCalculus Unit 2: Exponential and Logarithmic Functions Practice Exam

Assessment for Unit 2: Exponential and Logarithmic Functions

Estimated Time: 79 minutes

Exam Details

Questions: 63 total (59 multiple choice, 4 short answer)
Time: 79 minutes (recommended)
Topics Covered: Unit 2: Exponential and Logarithmic Functions
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (59 Questions)

Select the one best answer for each question.

1. A landscaping company plants a tree that is currently 150 cm tall. The tree is expected to grow by 12 cm each year. If $h_n$ represents the height of the tree in centimeters after $n$ years, which of the following equations best models the height of the tree?

(A)$h_n = 150(1.12)^n$(B)$h_n = 150 + 12n$(C)$h_n = 12 + 150n$(D)$h_n = 150(12)^n$

2. A new car is purchased for 30,000 dollars. The value of the car is expected to depreciate by 15% each year. If $v_n$ represents the value of the car, in dollars, after $n$ years, which of the following equations best models the value of the car?

(A)$v_n = 30000 - 4500n$(B)$v_n = 30000(0.15)^n$(C)$v_n = 30000(0.85)^n$(D)$v_n = 30000(1.15)^n$

3. The graph of the sequence $a_n$ consists of the discrete points $(0, 3)$, $(1, 6)$, $(2, 12)$, and $(3, 24)$. Which of the following defines the general term $a_n$ for $n \ge 0$?

(A)$a_n = 3 + 3n$(B)$a_n = 3(2)^n$(C)$a_n = 3(3)^n$(D)$a_n = 2(3)^n$

4. Consider a sequence where the initial term is $a_0 = 8$ and the first term is $a_1 = 12$. Let $A$ be the value of the term $a_2$ if the sequence is arithmetic, and let $G$ be the value of the term $a_2$ if the sequence is geometric. What is the value of $G - A$?

(A)0(B)2(C)4(D)6

5. The function $g$ is defined by $g(x) = 5^x$. The function $f$ is the inverse of $g$. Which of the following tables best represents selected values for the function $f$?

(A)x: -1, 0, 1, 2 $\\ $f(x): 0.2, 1, 5, 25(B)x: 0.2, 1, 5, 25 $\\ $f(x): -1, 0, 1, 2(C)x: -1, 0, 1, 2 $\\ $f(x): -5, 0, 5, 10(D)x: 0.2, 1, 5, 25 $\\ $f(x): 5, 1, 0.2, 0.04

6. The graph of the function $h$ is given by $h(x) = 3^x$. The function $k$ is the inverse of $h$. Which of the following statements is true regarding the graph of $k$?

(A)The graph of $k$ is the reflection of the graph of $h$ over the x-axis.(B)The graph of $k$ is the reflection of the graph of $h$ over the y-axis.(C)The graph of $k$ is the reflection of the graph of $h$ over the line $y=x$.(D)The graph of $k$ is the reflection of the graph of $h$ over the line $y=-x$.

7. Values of a function $f$ are given in the table below.$\n\n$| x | 2 | 4 | 8 | 16 |$\n$|---|---|---|---|----|$\n$| f(x) | 1 | 2 | 3 | 4 |$\n\nWhich $of the following statements best describes the relationship between the inputs and outputs of $f$, confirming that $f$ models logarithmic growth?

(A)As the input values increase additively by 2, the output values increase additively by 1.(B)As the input values increase multiplicatively by a factor of 2, the output values increase additively by 1.(C)As the input values increase multiplicatively by a factor of 2, the output values increase multiplicatively by a factor of 2.(D)As the input values increase additively by 2, the output values increase multiplicatively by a factor of 2.

8. The function $f$ is given by $f(x) = e^x$. If the function $g$ is the inverse of $f$, what is the value of $g(e^{-3})$?

(A)-3(B)1/e^3(C)e^3(D)undefined

9. The function $f$ is defined by $f(x) = \log_b x$, where $b$ is a constant such that $0 < b < 1$. Which of the following statements is true about the graph of $f$?

(A)The graph of $f$ is increasing and concave up.(B)The graph of $f$ is increasing and concave down.(C)The graph of $f$ is decreasing and concave up.(D)The graph of $f$ is decreasing and concave down.

10. The table below gives values for a function $g$ for selected values of $x$. | $x$ | 2 | 6 | 18 | 54 | | :--- | :---: | :---: | :---: | :---: | | $g(x)$ | 3 | 5 | 7 | 9 | Which of the following statements best explains why $g$ could be a logarithmic function?

(A)For every interval of $x$ of equal length, the value of $g(x)$ increases by a constant factor.(B)For every interval of $g(x)$ of equal length, the values of $x$ change by a constant factor.(C)The average rate of change of $g$ is constant over the intervals shown.(D)The value of $g(x)$ is always positive and increasing.

11. Let $f$ be the function given by $f(x) = \ln x$. Let $R_A$ be the average rate of change of $f$ over the interval $[2, 4]$, and let $R_B$ be the average rate of change of $f$ over the interval $[10, 12]$. Which of the following correctly compares $R_A$ and $R_B$ and provides a valid justification?

(A)$R_A > R_B > 0$, because the graph of $f$ is increasing and concave down.(B)$R_B > R_A > 0$, because the graph of $f$ is increasing and concave up.(C)$R_A < R_B < 0$, because the graph of $f$ is decreasing and concave down.(D)$R_A = R_B$, because the function $f$ grows at a constant rate.

12. Consider the function $h(x) = \log_{10}(x)$. Which of the following describes the end behavior and asymptotic behavior of the graph of $h$?

(A)$\lim_{x \to \infty} h(x) = 0$ and the graph has a horizontal asymptote at $y=0$.(B)$\lim_{x \to \infty} h(x) = \infty$ and the graph has a horizontal asymptote at $y=10$.(C)$\lim_{x \to \infty} h(x) = \infty$ and the graph has a vertical asymptote at $x=0$.(D)$\lim_{x \to \infty} h(x) = 1$ and the graph has a vertical asymptote at $x=0$.

13. [Skill: 1.B | Topic: 2.12] A student is using a calculator that has only the natural logarithm function, $\ln$. To evaluate the expression $\log_{5}(x^3)$ for positive values of $x$, the student rewrites it using logarithm properties. Which of the following expressions is equivalent to $\log_{5}(x^3)$ for $x>0$?

(A)$\dfrac{3\ln x}{\ln 5}$(B)$\dfrac{\ln (x^3)}{3\ln 5}$(C)$\dfrac{3\ln 5}{\ln x}$(D)$\dfrac{\ln x}{3\ln 5}$

14. Which of the following functions is equivalent to $g(x) = \log_{16}(x)$ for all $x > 0$?

(A)$h(x) = 4\log_2 x$(B)$k(x) = 2\log_4 x$(C)$p(x) = \frac{1}{2}\log_4 x$(D)$r(x) = \log_4(x^2)$

15. If $u = \ln x$ and $v = \ln y$, which of the following expressions is equivalent to $\ln \left( \frac{e^3 \sqrt{x}}{y^2} \right)$?

(A)$3 + \frac{1}{2}u - 2v$(B)$3 + 2u - \frac{1}{2}v$(C)$e^3 + \frac{1}{2}u - 2v$(D)$3 + \sqrt{u} - v^2$

16. Solve the equation $\log_{2}(x) + \log_{2}(x-2) = 3$ for $x$.

(A)x = 4 only(B)x = -2 only(C)x = 4 and x = -2(D)x = 3 and x = 5

17. The function $f$ is given by $f(x) = 2\log_{3}(x-1) + 4$. Which of the following defines the inverse function $f^{-1}$?

(A)f^{-1}(x) = 3^{$\frac{x-4}{2}$} + 1(B)f^{-1}(x) = 3^{$\frac{x}{2} $- 2} - 1(C)f^{-1}(x) = $\frac{3^{x-4}}{2} $+ 1(D)f^{-1}(x) = 3^{2(x-4)} + 1

18. What are all values of $x$ for which $\left(\frac{1}{2}\right)^{x+2} \geq 4^{2x-1}$?

(A)x $\leq $0(B)x $\geq $0(C)x $\leq \frac{4}{5}$(D)x $\geq \frac{4}{5}$

19. The table below shows values for the function $f$ at selected values of $x$. | $x$ | $2$ | $6$ | $18$ | $54$ | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | $5$ | $8$ | $11$ | $14$ | Which of the following function models is most appropriate for the relationship shown in the table?

(A)$f(x) = a + b \cdot x$(B)$f(x) = a \cdot b^x$(C)$f(x) = a + b \ln x$(D)$f(x) = a \cdot \ln(x+b)$

20. A researcher is modeling the height of a plant, $h(t)$, in centimeters, as a function of time $t$ in days using the function $h(t) = a + b \ln t$. The researcher collects the following data: at $t=1$ day, the height is $12$ cm, and at $t=e^2$ days, the height is $18$ cm. Based on this model, what is the predicted height of the plant at $t=e^5$ days?

(A)21 cm(B)27 cm(C)30 cm(D)42 cm

21. Students are analyzing a set of bivariate data $(x, y)$ to determine the best function model. They produce four different scatterplots using transformations of the data. Which of the following scatterplot descriptions indicates that the data should be modeled by a logarithmic function of the form $y = a + b \ln x$?

(A)A plot of $y$ versus $x$ shows a linear pattern.(B)A plot of $\ln y$ versus $x$ shows a linear pattern.(C)A plot of $y$ versus $\ln x$ shows a linear pattern.(D)A plot of $\ln y$ versus $\ln x$ shows a linear pattern.

22. [Skill: 2.B | Topic: 2.15] The relationship between variables $x$ and $y$ is analyzed using a semi-log plot where the vertical axis is logarithmically scaled with base 10. The trend line for the data on this semi-log plot is given by the equation $\log_{10} y = 0.4x + 1.2$. Which of the following exponential functions best models the relationship between $x$ and $y$?

(A)y = 1.2(0.4)^x(B)y = 10^{0.4}(10^{1.2})^x(C)y = 10^{1.2}(10^{0.4})^x(D)y = 1.2 + 0.4^x

23. [Skill: 3.B | Topic: 2.15] A researcher collects the following data for an independent variable $t$ and a dependent variable $P$: | $t$ | 0 | 2 | 4 | 6 | |---|---|---|---|---| | $P$ | 5.0 | 15.0 | 45.0 | 135.0 | The researcher wishes to determine if an exponential model is appropriate for this data set. Which of the following plots would result in a linear configuration of data points?

(A)A plot of P versus t(B)A plot of P versus log(t)(C)A plot of log(P) versus t(D)A plot of log(P) versus log(t)

24. [Skill: 2.A | Topic: 2.15] A semi-log plot displays the relationship between $x$ and the natural logarithm of $y$, denoted as $\ln y$. The graph shows a straight line passing through the points $(0, 3)$ and $(2, 4)$. Which of the following functions correctly models the relationship between $y$ and $x$?

(A)y = e^3 $\cdot $(e^{0.5})^x(B)y = 3 $\cdot $(0.5)^x(C)y = e^3 $\cdot $(e^2)^x(D)y = 3 $\cdot $e^{0.5x}

Refer to the figure below.

25. The table below gives values for a function $g$ at selected values of $x$. Which of the following statements is true regarding the type of function that best models $g$?

(A)$g$ is linear because the ratio of consecutive output values is constant for equal-length input intervals.(B)$g$ is linear because the difference of consecutive output values is constant for equal-length input intervals.(C)$g$ is exponential because the ratio of consecutive output values is constant for equal-length input intervals.(D)$g$ is exponential because the difference of consecutive output values is constant for equal-length input intervals.

26. A quantity is measured at time $t=0$ to be 200. At time $t=3$, the quantity is measured to be 160. Let $L(t)$ be a linear model for the quantity and let $E(t)$ be an exponential model for the quantity. Which of the following compares the predicted values of the quantity at $t=6$ using the two models?

(A)$L(6) < E(6)$(B)$L(6) = E(6)$(C)$L(6) > E(6)$(D)The relationship cannot be determined without the explicit equations.

27. The table below shows values for a function f at selected values of x . | x | 2 | 4 | 6 | 8 | |---|---|---|---|---| | f(x) | 12 | 18 | 27 | 40.5 | Which of the following claims about f is true?

(A) f is best modeled by a linear function because the rate of change is constant.(B) f is best modeled by a linear function because the differences in output values are increasing.(C) f is best modeled by an exponential function because the ratio of output values over equal input intervals is constant.(D) f is best modeled by an exponential function because the graph of f is concave up.

28. Let g be the function defined by g(x) = -3(0.4)^x . Which of the following describes the behavior of the graph of g in the xy -plane?

(A)Increasing and concave up(B)Increasing and concave down(C)Decreasing and concave up(D)Decreasing and concave down

29. A biological culture contains a population of bacteria, P , that changes with time, t , measured in hours. The population triples every 5 hours. If the initial population is P_0 , which of the following functions models the population P(t) ?

(A) P(t) = P_0 (3)^{5t} (B) P(t) = P_0 (3)^{$\frac{t}{5}$} (C) P(t) = P_0 $\left$($\frac{1}{3}\right$)^{5t} (D) P(t) = 3 P_0 (5)^t

30. The exponential function $f$ is defined by $f(x) = 8(2)^{x-3}$. Which of the following is an equivalent form of $f(x)$ that shows the function as a vertical dilation of the parent function $g(x) = 2^x$?

(A)$f(x) = (2)^x$(B)$f(x) = 5(2)^x$(C)$f(x) = 64(2)^x$(D)$f(x) = 8(2)^{-3x}$

31. The function $g$ is given by $g(x) = 3(16)^{3x/4}$. Which of the following is an equivalent form of $g(x)$?

(A)$3(8)^x$(B)$3(12)^x$(C)$3(64)^x$(D)$48^x$

32. A scientist models the population of a specific bacteria colony with the function $P(t) = 500(1.4)^t$, where $t$ represents the time in days. Which of the following functions $M$ models the population where $m$ represents the time in hours?

(A)$M(m) = 500(1.4)^{24m}$(B)$M(m) = 500(1.4^{24})^m$(C)$M(m) = 500(\frac{1.4}{24})^m$(D)$M(m) = 500(1.4^{1/24})^m$

33. The table below gives values for a function $f$ at selected values of $x$. Which of the following statements explains why $f$ could be best modeled by an exponential function? | $x$ | 2 | 4 | 6 | 8 | |---|---|---|---|---| | $f(x)$ | 16 | 24 | 36 | 54 |

(A)The rate of change of $f$ is constant over equal-length input-value intervals.(B)The value of $f$ increases by a constant amount for every 2-unit increase in $x$.(C)The ratio of consecutive values of $f$ is constant for equal-length input-value intervals.(D)The ratio of the value of $f$ to the value of $x$ is constant for all entries in the table.

34. A biologist models the population of a bacterial colony with the function $P(t) = 500(1.08)^{2t}$, where $t$ is measured in hours. Which of the following is the best interpretation of the parameter $1.08$ in the context of this model?

(A)The population increases by $8\%$ every hour.(B)The population increases by $8\%$ every half-hour.(C)The population increases by $16\%$ every hour.(D)The population increases by $108\%$ every hour.

35. The value of a vintage car, $V(t)$, in thousands of dollars, is modeled by an exponential function of time $t$ in years. When $t=0$, the value of the car is 20 thousand dollars. When $t=3$, the value of the car is 25 thousand dollars. Which of the following expressions defines $V(t)$?

(A)$20(1.25)^t$(B)$20(1.25)^{t/3}$(C)$20(1.08)^{t}$(D)$20 + \frac{5}{3}t$

36. A researcher collects data on the population of a specific bacteria culture over time. The table below shows the population, $P(t)$, in thousands, at various times $t$ in hours. | $t$ (hours) | $P(t)$ (thousands) | | :---: | :---: | | 0 | 12.00 | | 2 | 18.00 | | 4 | 27.00 | | 6 | 40.50 | Based on the data in the table, which of the following function types best models the relationship between $t$ and $P(t)$, and why?

(A)Linear, because the rate of change of the population is increasing at a constant rate.(B)Linear, because the difference between consecutive population values is constant.(C)Quadratic, because the second differences of the population values are constant.(D)Exponential, because the ratio of consecutive population values for equal intervals of time is constant.

37. A student uses a regression calculator to fit a linear model, $y = ax + b$, to a set of data points $(x, y)$. To validate whether the linear model is appropriate, the student generates a residual plot. The residual plot shows a distinct U-shaped pattern, with positive residuals for low and high $x$-values and negative residuals for middle $x$-values. Which of the following is the correct conclusion based on this residual plot?

(A)The linear model is appropriate because the sum of the residuals is approximately zero.(B)The linear model is appropriate because there is an equal number of points above and below the $x$-axis.(C)The linear model is not appropriate because the pattern in the residual plot suggests a non-linear relationship between the variables.(D)The linear model is not appropriate because the residuals are not all equal to zero.

38. The table below gives values for the variables $x$ and $y$. | $x$ | 1 | 2 | 3 | 4 | 5 | |:---:|:---:|:---:|:---:|:---:|:---:| | $y$ | 4.2 | 7.8 | 11.4 | 15.0 | 18.6 | A student is deciding between a linear and an exponential model for the data. Which of the following statements provides the strongest evidence for selecting a linear model?

(A)The ratio of consecutive $y$-values decreases as $x$ increases.(B)The average rate of change between consecutive points is constant.(C)The second differences of the $y$-values are approximately constant and non-zero.(D)The $y$-values are always increasing as $x$ increases.

39. The function $f$ is given by $f(x) = 5 - e^{x+3}$. Which of the following defines the inverse function $f^{-1}$ ?

(A)f^{-1}(x) = $\ln$(5 - x) - 3(B)f^{-1}(x) = $\ln$(x - 5) - 3(C)f^{-1}(x) = $\ln$(5 - x) + 3(D)f^{-1}(x) = 3 - $\ln$(5 - x)

40. The function $g$ is defined by $g(x) = 2\log_4(x - 3) + 1$. What are the domain and range of the inverse function, $g^{-1}$ ?

(A)Domain: (3, $\infty$); Range: (-$\infty$, $\infty$)(B)Domain: (-$\infty$, $\infty$); Range: (3, $\infty$)(C)Domain: (-$\infty$, $\infty$); Range: (1, $\infty$)(D)Domain: (1, $\infty$); Range: (3, $\infty$)

41. The population of a colony of bacteria $t$ hours after an experiment begins is modeled by the function $P(t) = 50(2)^{t/4}$. Which of the following is true about the value and interpretation of $P^{-1}(200)$ ?

(A)P^{-1}(200) = 8, representing that it takes 8 hours for the population to reach 200 bacteria.(B)P^{-1}(200) = 8, representing that the population is 8 bacteria after 200 hours.(C)P^{-1}(200) = 16, representing that it takes 16 hours for the population to reach 200 bacteria.(D)P^{-1}(200) = 16, representing that the population is 16 bacteria after 200 hours.

42. Evaluate the following expression: $$\log_4(32) + \log_9(27)$$

(A)3(B)3.5(C)4(D)5

43. Values $A$ and $B$ are plotted on a logarithmic scale with a base of 3. If the position of $B$ is 4 units greater than the position of $A$ on this scale, which of the following statements is true regarding the values of $A$ and $B$?

(A)The value of $B$ is 12 times the value of $A$.(B)The value of $B$ is 64 times the value of $A$.(C)The value of $B$ is 81 times the value of $A$.(D)The value of $B$ is 243 times the value of $A$.

44. Consider the following logarithmic values: $$x = \log_3(20)$$ $$y = \log_4(20)$$ $$z = \log_5(20)$$ Which of the following correctly orders $x$, $y$, and $z$?

(A)$x < y < z$(B)$z < y < x$(C)$y < z < x$(D)$z < x < y$

45. [Skill: 2.B | Topic: 2.7] Let $f$ and $g$ be functions defined by $f(x) = e^x$ and $g(x) = \ln(x^4)$ for $x > 0$. Which of the following is an analytic representation of the composite function $h(x) = f(g(x))$?

(A)$h(x) = x^4$(B)$h(x) = 4x$(C)$h(x) = 4 \ln(e^x)$(D)$h(x) = e^x \ln(x^4)$

46. [Skill: 1.C | Topic: 2.7] The functions $f$ and $g$ are defined by $f(x) = 2x - 3$ and $g(x) = \ln(x + 5)$. Which of the following is an analytic representation of $h(x) = f(g(x))$?

(A)h(x) = 2$\ln$(x + 5) - 3(B)h(x) = $\ln$(2x + 2)(C)h(x) = $\ln$(2x - 3) + 5(D)h(x) = (2x - 3)$\ln$(x + 5)

47. [Skill: 2.B | Topic: 2.2] The table below shows values for a linear function $f$ and an exponential function $g$ for specific values of $x$. | $x$ | $f(x)$ | $g(x)$ | | :--- | :--- | :--- | | 0 | 4 | 4 | | 2 | 16 | 16 | Which of the following statements is true regarding the values of $f(3)$ and $g(3)$? (A) $f(3) = 22$ and $g(3) = 32$, so $g(3) > f(3)$. (B) $f(3) = 22$ and $g(3) = 22$, so $f(3) = g(3)$. (C) $f(3) = 28$ and $g(3) = 32$, so $g(3) > f(3)$. (D) $f(3) = 22$ and $g(3) = 64$, so $g(3) > f(3)$.

(A)f(3) = 22 and g(3) = 32, so g(3) > f(3).(B)f(3) = 22 and g(3) = 22, so f(3) = g(3).(C)f(3) = 28 and g(3) = 32, so g(3) > f(3).(D)f(3) = 22 and g(3) = 64, so g(3) > f(3).

48. Let $f$ and $g$ be functions defined by $f(x) = \frac{1}{x}$ and $g(x) = \ln(x - 3)$. Which of the following defines the domain of the composite function $h$ defined by $h(x) = f(g(x))$?

(A)$(3, \infty)$(B)$(3, 4) \cup (4, \infty)$(C)$(4, \infty)$(D)$(-\infty, 4) \cup (4, \infty)$

49. Let $f$ be the function defined by $f(x) = \log_2(x)$. The values of the function $g$ are given in the table below. | $x$ | $2$ | $4$ | $8$ | $16$ | | :---: | :---: | :---: | :---: | :---: | | $g(x)$ | $8$ | $32$ | $4$ | $2$ | What is the value of $(f \circ g)(4)$?

(A)3(B)5(C)8(D)32

Refer to the figure below.

50. Selected values of the function f are given in the table below. The graph of the function g consists of a line segment connecting the points (0, 4) and (4, 0) . **Table for f(x) :** | x | f(x) | |---|---| | 2 | 4 | | 4 | 16 | | 8 | 64 | What is the value of (g $\circ $f)(2) ?

(A)0(B)2(C)4(D)8

51. [Skill: 1.B | Topic: 2.7] Let $f(x) = \log_2(x)$ and let $g(x) = 8 - 2^x$. Which of the following defines the domain of the composite function $h(x) = f(g(x))$ ?

(A)$(-\infty, 3)$(B)$(-\infty, 3]$(C)$(3, \infty)$(D)$(-\infty, \infty)$

52. Let $f(x) = e^{2x}$ and $g(x) = \ln(x)$. Which of the following defines the composite function $h(x) = f(g(x))$ ?

(A)$h(x) = 2x$ for $x > 0$(B)$h(x) = x^2$ for $x > 0$(C)$h(x) = 2x$ for all real values of $x$(D)$h(x) = x^2$ for all real values of $x$

53. [Skill: 1.B | Topic: 2.7] Let $f$ and $g$ be functions defined by $f(x) = \sqrt{x}$ and $g(x) = 2 - \log_3(x)$. What is the domain of the composite function $h$ defined by $h(x) = f(g(x))$?

(A)(0, 9](B)[9, $\infty$)(C)(0, $\infty$)(D)(-$\infty$, 9]

54. Let $f$ and $g$ be functions defined by $f(x) = e^x$ and $g(x) = \ln(2x - 6)$. Which of the following defines the function $h$ given by $h(x) = (f \circ g)(x)$ ?

(A)$h(x) = 2x - 6$ for all real values of $x$(B)$h(x) = 2x - 6$ for $x > 3$(C)$h(x) = 2x - 6$ for $x > 0$(D)$h(x) = \ln(2e^x - 6)$ for $x > \ln(3)$

55. [Skill: 1.B | Topic: 2.7] Let $f(x) = \ln(x)$ and $g(x) = x^2 - 4x - 5$. Which of the following describes the domain of the composite function $h(x) = f(g(x))$?

(A)$(-1, 5)$(B)$(5, \infty)$(C)$(-\infty, -1) \cup (5, \infty)$(D)$(-\infty, -1] \cup [5, \infty)$

56. Let $f$ and $g$ be functions defined by $f(x) = \log_2(x)$ and $g(x) = x^2 - 4x - 12$. Which of the following describes the domain of the composite function $h$ defined by $h(x) = f(g(x))$?

(A)(-2, 6)(B)(-$\infty$, -2) $\cup $(6, $\infty$)(C)(6, $\infty$)(D)(0, $\infty$)

57. Let $f$ and $g$ be functions defined by $f(x) = 2 \cdot 3^{x+1}$ and $g(x) = \log_3(x-1)$. What is the value of $f(g(10))$?

(A)18(B)54(C)162(D)216

58. The function $f$ is given by the graph below, and the function $g$ is defined by the table of values. [Image Cue]: A graph of a logarithmic function $f(x)$ plotted on the xy-plane. The graph passes through the points $(1, 0)$, $(2, 1)$, $(4, 2)$, and $(8, 3)$. **Table for $g(x)$** | $x$ | $g(x)$ | |---|---| | 1 | 8 | | 2 | 4 | | 3 | 2 | | 4 | 1 | What is the value of $g(f(4))$?

(A)0(B)1(C)2(D)4

59. [Skill: 2A | Topic: 2.7] A sound engineer uses two functions to model a signal-processing step. The functions are defined by - $f(x)=\ln(x)$ - $g(x)=3^x$ Which of the following correctly states the value of $(f\circ g)(-2)$ and whether $(g\circ f)(-2)$ is defined for real numbers?

(A)$(f\circ g)(-2)=-2\ln(3)$, and $(g\circ f)(-2)$ is not defined for real numbers(B)$(f\circ g)(-2)=\ln(-2)$, and $(g\circ f)(-2)=3^{-2}$(C)$(f\circ g)(-2)=\ln(9)$, and $(g\circ f)(-2)$ is not defined for real numbers(D)$(f\circ g)(-2)$ is not defined for real numbers, and $(g\circ f)(-2)=-2\ln(3)$

B. Short Answer Questions (4 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 60:

Question 61:

Question 62:

Question 63:

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