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Questions and Answers
Does the statement 'A population of bacteria decreases by a factor of 4 every 12 hours' represent an exponential function?
Does the statement 'A population of bacteria decreases by a factor of 4 every 12 hours' represent an exponential function?
What type of growth does the equation $f(x) = 18.4 imes 1.025^t$ represent?
What type of growth does the equation $f(x) = 18.4 imes 1.025^t$ represent?
Which statement about the radioactive isotope with a half-life of 15 hours is correct?
Which statement about the radioactive isotope with a half-life of 15 hours is correct?
Which function represents exponential decay among the following?
Which function represents exponential decay among the following?
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What can be inferred about the function represented by the table where the values of $f(x)$ are $-1, -4, -16, -64$?
What can be inferred about the function represented by the table where the values of $f(x)$ are $-1, -4, -16, -64$?
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Which of the following values indicates that a function is linear?
Which of the following values indicates that a function is linear?
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Which of the following equations illustrates neither exponential growth nor decay?
Which of the following equations illustrates neither exponential growth nor decay?
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What is true about the function $k(x) = 3.5, 7, 14, 28, 56$ observed in a table?
What is true about the function $k(x) = 3.5, 7, 14, 28, 56$ observed in a table?
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What is the function that represents the volume of the balloon after t seconds if it inflates at a constant rate of 40 cubic millimeters per second?
What is the function that represents the volume of the balloon after t seconds if it inflates at a constant rate of 40 cubic millimeters per second?
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If the radius of a circular puddle is given by $r(t) = 2t + 3$, what is the area function expressed in terms of time?
If the radius of a circular puddle is given by $r(t) = 2t + 3$, what is the area function expressed in terms of time?
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What is the radius of the balloon after 3 seconds if the relationship is defined by $V = 3 ext{π}r^3$?
What is the radius of the balloon after 3 seconds if the relationship is defined by $V = 3 ext{π}r^3$?
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What is the composite function that represents the number of Mexican pesos equivalent to 750 Japanese yen?
What is the composite function that represents the number of Mexican pesos equivalent to 750 Japanese yen?
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After 12 minutes, what is the area of the puddle if the radius at that time is given by $r(t) = 2(12) + 3$?
After 12 minutes, what is the area of the puddle if the radius at that time is given by $r(t) = 2(12) + 3$?
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What does the function $a(x) = ext{π}x^2$ represent in the context of circular ripples?
What does the function $a(x) = ext{π}x^2$ represent in the context of circular ripples?
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How do you find the radius after t seconds for a balloon that inflates at a constant rate?
How do you find the radius after t seconds for a balloon that inflates at a constant rate?
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If the conversion from Japanese yen to US dollars is given by $f(x) = 0.0085x$, what is the equivalent of 1000 Japanese yen?
If the conversion from Japanese yen to US dollars is given by $f(x) = 0.0085x$, what is the equivalent of 1000 Japanese yen?
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What is the end behavior of the function $f(x) = 4e^x$ as $x$ approaches $-eta$?
What is the end behavior of the function $f(x) = 4e^x$ as $x$ approaches $-eta$?
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What is the equation of the horizontal asymptote for the function $g(x) = 3 imes 2^{-x} + c$?
What is the equation of the horizontal asymptote for the function $g(x) = 3 imes 2^{-x} + c$?
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What value does the function $g(x) = 3 imes 2^{-x} + c$ approach as $x$ approaches infinity?
What value does the function $g(x) = 3 imes 2^{-x} + c$ approach as $x$ approaches infinity?
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What is the proper limit notation for the end behavior of the function $F(x) = a b^x + c$ as $x$ approaches $-eta$?
What is the proper limit notation for the end behavior of the function $F(x) = a b^x + c$ as $x$ approaches $-eta$?
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What can be concluded about the value of $a$ in the function $F(x) = a b^x + c$ when the function approaches $c$?
What can be concluded about the value of $a$ in the function $F(x) = a b^x + c$ when the function approaches $c$?
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For the function $f(x)$ represented in the table, as $x$ approaches 1, what is the function value $F(1)$?
For the function $f(x)$ represented in the table, as $x$ approaches 1, what is the function value $F(1)$?
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Which of the following describes the range of the function $G(x) = a b^{-x} + c$?
Which of the following describes the range of the function $G(x) = a b^{-x} + c$?
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In the given table for the function $F(x)$, what behavior is observed as $x$ approaches -8?
In the given table for the function $F(x)$, what behavior is observed as $x$ approaches -8?
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What is the domain of the function $f(x) = log(x - 3)$?
What is the domain of the function $f(x) = log(x - 3)$?
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What can be said about the left-end behavior of the function $h(x) = 2 log(x) - 3$?
What can be said about the left-end behavior of the function $h(x) = 2 log(x) - 3$?
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What does the asymptote of the function $g(x) = ln(4 - x)$ represent?
What does the asymptote of the function $g(x) = ln(4 - x)$ represent?
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For the function $f(x) = -ln(x) + 3 - 4$, what is the correct interpretation of the range?
For the function $f(x) = -ln(x) + 3 - 4$, what is the correct interpretation of the range?
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Which behavior describes the function $f(x) = log(x - 3)$ as $x$ approaches infinity?
Which behavior describes the function $f(x) = log(x - 3)$ as $x$ approaches infinity?
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What type of behavior is exhibited by the function defined by the values in the table: $x = 1, 2, 3, 4, 5$ and $f(x) = -1, 1, 5, 13, 29$?
What type of behavior is exhibited by the function defined by the values in the table: $x = 1, 2, 3, 4, 5$ and $f(x) = -1, 1, 5, 13, 29$?
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Given the table with $x = 2, 4, 8, 16, 32$ and $g(x) = 1, 2, 3, 4, 5$, which behavior does it illustrate?
Given the table with $x = 2, 4, 8, 16, 32$ and $g(x) = 1, 2, 3, 4, 5$, which behavior does it illustrate?
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What is the right-end behavior of the function $h(x) = 2 log(x) - 3$?
What is the right-end behavior of the function $h(x) = 2 log(x) - 3$?
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What is the explicit formula for the sequence of loan balances, where the balances are 960, 835, 710, and 585?
What is the explicit formula for the sequence of loan balances, where the balances are 960, 835, 710, and 585?
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If Cameron's loan balance is 585 after 4 months, how much did he initially borrow?
If Cameron's loan balance is 585 after 4 months, how much did he initially borrow?
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How many months will it take for Cameron to fully repay his loan if he makes equal monthly payments?
How many months will it take for Cameron to fully repay his loan if he makes equal monthly payments?
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How much will your total salary exceed $500,000 given a starting salary of $45,000 and an annual raise of $2,500?
How much will your total salary exceed $500,000 given a starting salary of $45,000 and an annual raise of $2,500?
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What will be your total gross salary at the end of the first year with a starting salary of $45,000?
What will be your total gross salary at the end of the first year with a starting salary of $45,000?
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In the geometric sequence starting with {1250, 250, 50, 10, ...}, what is the common ratio?
In the geometric sequence starting with {1250, 250, 50, 10, ...}, what is the common ratio?
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What explicit rule models the sequence where the second term is 4 and the common ratio is 3?
What explicit rule models the sequence where the second term is 4 and the common ratio is 3?
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What is the next term in the sequence {-2, -8, -32, ...}?
What is the next term in the sequence {-2, -8, -32, ...}?
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What is the vertical asymptote of the function 𝑓(𝑥) = ln(𝑥 + 3)?
What is the vertical asymptote of the function 𝑓(𝑥) = ln(𝑥 + 3)?
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Which of the following statements about the function 𝐹(𝑥) = log₃(2𝑥 - 1) - 4 is correct?
Which of the following statements about the function 𝐹(𝑥) = log₃(2𝑥 - 1) - 4 is correct?
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What is the domain of the function 𝐺(𝑥) = 4 - 2 log₂(3 - 𝑥)?
What is the domain of the function 𝐺(𝑥) = 4 - 2 log₂(3 - 𝑥)?
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As 𝑥 approaches negative infinity for the function 𝐺(𝑥), what is the behavior of 𝐺?
As 𝑥 approaches negative infinity for the function 𝐺(𝑥), what is the behavior of 𝐺?
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What can be determined about the concavity of the function 𝑓(𝑥) using the values from the given table?
What can be determined about the concavity of the function 𝑓(𝑥) using the values from the given table?
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For the function 𝐻(𝑥) = log₂(𝑥 + 1), what is the behavior as 𝑥 approaches 0 from the right?
For the function 𝐻(𝑥) = log₂(𝑥 + 1), what is the behavior as 𝑥 approaches 0 from the right?
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What is the zero of the function 𝑓(𝑥) = 𝑙𝑜𝑔₂(𝑥) + 2?
What is the zero of the function 𝑓(𝑥) = 𝑙𝑜𝑔₂(𝑥) + 2?
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When analyzing the function 𝑘(𝑥) = -log₃(−𝑥), what happens as 𝑥 approaches 0 from the left?
When analyzing the function 𝑘(𝑥) = -log₃(−𝑥), what happens as 𝑥 approaches 0 from the left?
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Study Notes
Arithmetic and Geometric Sequences
- Arithmetic sequences have a constant difference between consecutive terms.
- Geometric sequences have a constant ratio between consecutive terms.
Finding the nth term of an arithmetic sequence.
- Use the formula: an = a₁ + (n - 1)d
- an = the nth term
- a₁ = the first term
- n = the term number
- d = the common difference
Finding the nth term of a geometric sequence.
- Use the formula: an = a₁⋅ r^(n-1)
- an = the nth term
- a₁ = the first term
- n = the term number
- r = the common ratio
Problems with Arithmetic and Geometric Sequences
- Problems 1-4: Determine if the sequence is arithmetic. Find the common difference if applicable.
- Problems 5-8: Find the first five terms of a sequence.
- Problems 9-12: Find the 28th term of a sequence.
- Problems 13-16: Find the general rule for a sequence given the 13th term (or similar) and the common difference.
- Problems 17-21: Solve word problems.
Geometric Sequences: Common Ratio
- Common ratio (r) is the constant ratio between consecutive terms in a geometric sequence.
- Calculate the common ratio by dividing any term by the previous term.
Problems with Geometric Sequences:
- Problems 1-4: Determine if the sequence is geometric. Find common ratio if applicable.
- Problems 5-8: Find the explicit formula (nth term) for given sequences.
- Problems 9-12: Determine the missing term of the sequence.
Linear and Exponential Functions
-
Linear functions have a constant rate of change (slope).
-
Explicit rule for a linear function: f(n) = mn + b
- m = the slope, or rate of change
- b = the y-intercept, or initial value
-
Exponential functions have a constant multiplier (growth factor).
-
Explicit rule for an exponential function: f(n)=ab^n
- a = the initial value
- b = the growth or decay factor
Problems with Linear Functions
- Problems 1-2: Use the graph to complete the table, find the difference and write an explicit rule for a linear function.
Problems with Exponential Functions
- Problems 1-2: Use the graph to complete the table, find the ratio, and write an explicit rule for an exponential function.
Understanding Sequences
- Identify patterns and relationships within a sequence.
- Use formulas and rules to calculate terms in a sequence.
Word Problems
- Translate word problems into mathematical equations and solutions.
Logs/Exponents
- Understand logarithmic and exponential expressions
Studying That Suits You
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Description
This quiz explores various concepts related to exponential functions, including population decay models, growth equations, and characteristics of exponential behavior. You'll analyze functions based on given data and determine their growth or decay properties. Perfect for students looking to understand the fundamentals of exponential functions in mathematics.
