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Questions and Answers
Which characteristic is essential in defining a function?
Which characteristic is essential in defining a function?
What impact does compounding frequency have on compound interest?
What impact does compounding frequency have on compound interest?
What is a key feature of a piecewise function?
What is a key feature of a piecewise function?
How does simple interest differ from compound interest in terms of calculation?
How does simple interest differ from compound interest in terms of calculation?
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Which of the following statements about operations on functions is true?
Which of the following statements about operations on functions is true?
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What defines a simple annuity?
What defines a simple annuity?
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In the context of annuities, what does a general annuity allow?
In the context of annuities, what does a general annuity allow?
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What symbol is commonly used to denote a function?
What symbol is commonly used to denote a function?
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What will happen if a piecewise function is evaluated at a value falling between specified intervals?
What will happen if a piecewise function is evaluated at a value falling between specified intervals?
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When dealing with compound interest, what does 'n' represent in the formula A = P(1 + r/n)^(nt)?
When dealing with compound interest, what does 'n' represent in the formula A = P(1 + r/n)^(nt)?
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Study Notes
Functions
- Functions represent a relationship between inputs (domain) and outputs (range). Each input maps to exactly one output.
- Notation: y = f(x) (y is a function of x)
- Types of functions: linear, quadratic, exponential, polynomial, trigonometric, logarithmic.
- Key features of functions: domain, range, intercepts (x and y), increasing/decreasing intervals, maximum/minimum values, asymptotes.
Compound Interest
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Compound interest calculates interest earned not only on the principal amount but also on accumulated interest from previous periods. This results in faster growth compared to simple interest.
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Formula: A = P(1 + r/n)^(nt), where:
- A = the future value of the investment/loan, including interest
- P = the principal investment amount (the initial deposit or loan amount)
- r = the annual interest rate (decimal)
- n = the number of times that interest is compounded per year
- t = the number of years the money is invested or borrowed for.
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Understanding compounding frequency (annually, semiannually, quarterly, monthly, daily) is crucial for calculating compound interest accurately.
Piecewise and Evaluation
- A piecewise function is defined by multiple sub-functions, each applying over specific intervals of the input variable.
- To evaluate a piecewise function, determine which sub-function applies to the given input value.
Simple Interest
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Simple interest calculates interest only on the principal amount, not on accumulated interest.
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Formula: I = Prt, where:
- I = interest earned
- P = principal amount
- r = annual interest rate (decimal)
- t = time (in years)
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Simple interest is easier to calculate than compound interest, but results in slower growth.
Operations on Functions
- Functions can be added, subtracted, multiplied, and divided. Operations are performed on corresponding outputs for a given input.
- Examples include:
- f(x) + g(x) = (sum of functions)
- f(x) - g(x) = (difference of functions)
- f(x) * g(x) = (product of functions)
- f(x) / g(x) = (quotient of functions)
Simple Annuity
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A series of equal payments made at equal intervals over a specific period.
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Simple annuities typically involve fixed payments for a predetermined duration.
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Common examples include loan repayments and regular savings plans.
General Annuity
- General annuities allow for various payment situations. These could involve:
- payments that change over time,
- different payment intervals,
- varying interest rates during the annuity term.
- More complex calculations and formulas are needed to handle these scenarios compared to simple annuities.
- The time value of money plays a critical role in understanding and calculating both simple and general annuities, particularly when considering present value and future value.
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Description
Explore the essential concepts of functions and compound interest in this quiz. Learn about the relationship between inputs and outputs in functions, as well as the calculation of compound interest over time. Test your understanding of different types of functions and the formula for compound interest.
