Podcast
Questions and Answers
If a function is defined as $f(x)$, what notation represents its inverse?
If a function is defined as $f(x)$, what notation represents its inverse?
- $f^{-1}(x)$ (correct)
- $-f(x)$
- $f'(x)$
- $1/f(x)$
When finding the inverse of a function given by an equation, what is the first step?
When finding the inverse of a function given by an equation, what is the first step?
- Graph the function
- Replace $f(x)$ with 0
- Switch the places of x and y (correct)
- Solve for y
If a table of values represents a function, how do you find the table of values for the inverse function?
If a table of values represents a function, how do you find the table of values for the inverse function?
- Invert the sign of the x value and the y value
- Switch the x and y values in each ordered pair (correct)
- Multiply all x-values by -1
- Multiply all y-values by -1
What is the relationship between the graph of a function and the graph of its inverse?
What is the relationship between the graph of a function and the graph of its inverse?
In the general exponential model $y=ab^x$, what does 'a' represent?
In the general exponential model $y=ab^x$, what does 'a' represent?
What is the purpose of adding 1 to the rate (as a decimal) in the growth factor of an exponential model?
What is the purpose of adding 1 to the rate (as a decimal) in the growth factor of an exponential model?
What does 'n' represent in the compound interest formula?
What does 'n' represent in the compound interest formula?
In continuous compounding, which constant is used in the formula?
In continuous compounding, which constant is used in the formula?
How should a percentage rate be inputted into the exponential growth or decay formulas?
How should a percentage rate be inputted into the exponential growth or decay formulas?
In the context of population growth modeled by $y = ab^x$, why should the final answer be rounded to a whole number?
In the context of population growth modeled by $y = ab^x$, why should the final answer be rounded to a whole number?
Flashcards
Inverse of a Function
Inverse of a Function
Reverses the order of inputs and outputs of a function. Switch x and y.
Notation for Inverse
Notation for Inverse
Written as f⁻¹(x), reads as 'f inverse'. Switch x and y in the equation.
General Exponential Model
General Exponential Model
y = a * b^x, where 'a' is the initial value, 'b' is the growth/decay factor, and 'x' is the number of years.
Continuous Compound Interest
Continuous Compound Interest
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Graphing Inverses
Graphing Inverses
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X-intercept
X-intercept
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y
y
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y-intercept
y-intercept
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Study Notes
- The lesson covers inverse functions and exponential models
Inverse Functions
- Inverse functions reverse the order of outputs and inputs
- To find the inverse, switch x and y and solve for y
- The inverse of a function f is written as f⁻¹ (f inverse)
- When working with a table, switch the x and y values to find the inverse
Solving for y
- Subtract six from both sides to isolate the term with y
- To solve for y, divide everything by two, resulting in y = (1/2)x - 3
Graphing Inverses
- When graphing the original function and its inverse, they reflect over the line y = x
- Reflecting over y = x means the graph mirrors that line
Exponential Models
- Includes the general exponential model and compound interest models
- The general exponential model is y = a * bˣ, where:
- a is the initial value
- b is the growth factor
- x is the number of years
Growth Factor Example
- If a necklace costs $50 and increases in value by 2% per year, the equation is y = 50 * (1 + 0.02)ˣ
- Convert the percentage to a decimal (2% = 0.02) and add one for the growth factor
Compound Interest
- Compound interest formula: A = P(1 + r/ n)^(nt) where:
- P is the principal
- r is the rate (as a decimal)
- n is the number of times compounded per year
- t is the time in years
- For continuous interest, use the formula A = Pe^(rt)
Population Growth Example
- If Millville's population grows at 15% annually, the formula is y = a * bˣ where:
- a is the initial population
- b is 1 + 0.15 = 1.15
- x is the number of years
- Round the result to the nearest whole number since you can't have a decimal of a person
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Description
Explore inverse functions, which reverse the order of outputs and inputs, and learn how to find them by switching x and y values. Understand exponential models, including the general form y = a * b^x, where a is the initial value, b is the growth factor, and x is the number of years. See how graphs of functions and their inverses reflect over the line y = x.
