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What is the formula for the nth term of an arithmetic sequence?
$u_n = a + (n-1)d$, where 'a' is the first term and 'd' is the common difference.
What is the formula for the nth term of a geometric sequence?
$u_n = ar^{n-1}$, where 'a' is the first term and 'r' is the common ratio.
What is the general form of an exponential function?
$f(x) = ab^x$
How do you denote the inverse of a function f(x)?
$f^{-1}(x)$
What is the relationship between a function and its inverse?
$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
What is the formula for a composite function of f and g?
$(f \circ g)(x) = f(g(x))$
How to find the inverse of a function y=f(x)?
Swap x and y, then solve for y.
What is the logarithmic form of $b^y = x$?
$y = \log_b(x)$
What is the exponential form of $y = \log_b(x)$?
$b^y = x$
What is the change of base formula for logarithms?
$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$
Explain the concept of exponential growth.
Exponential growth occurs when a quantity increases proportionally to its current value, leading to rapid growth over time.
Explain the concept of exponential decay.
Exponential decay occurs when a quantity decreases proportionally to its current value, leading to rapid decline over time.
Explain the relationship between exponential and logarithmic functions.
Logarithmic functions are the inverses of exponential functions. They 'undo' each other.
Explain the significance of the base 'b' in an exponential function.
If b > 1, the function represents growth. If 0 < b < 1, the function represents decay.
Explain the significance of the base 'b' in a logarithmic function.
The base determines the rate at which the logarithm increases; it also defines the domain (x > 0).
Explain the concept of asymptotes in exponential and logarithmic functions.
Exponential functions have a horizontal asymptote, while logarithmic functions have a vertical asymptote. These are lines the graph approaches but never touches.
Explain the concept of domain and range for exponential functions.
The domain is all real numbers, and the range is all positive real numbers (if a > 0).
Explain the concept of domain and range for logarithmic functions.
The domain is all positive real numbers, and the range is all real numbers.
Explain how composite functions combine the transformations of individual functions.
The inner function's output becomes the input for the outer function, applying transformations sequentially.
Explain how inverse functions reflect across the line y=x.
Graphically, a function and its inverse are reflections of each other across the line y=x.
What is an exponential function?
A function of the form $f(x) = ab^x$, where a and b are constants, and b > 0 and not equal to 1.
What is a logarithmic function?
A function of the form $f(x) = \log_b(x)$, which is the inverse of an exponential function.
Define arithmetic sequence.
A sequence where each term differs by a constant amount (common difference).
Define geometric sequence.
A sequence where each term is multiplied by a constant amount (common ratio).
What is a composite function?
A function formed by combining two or more functions, where the output of one function becomes the input of another.
What are inverse functions?
Functions that 'undo' each other. If $f(g(x)) = x$ and $g(f(x)) = x$, then $f(x)$ and $g(x)$ are inverses.
What is a semi-log plot?
A graph that uses a logarithmic scale on one axis and a linear scale on the other.
Define the common difference in an arithmetic sequence.
The constant amount added to each term to get the next term.
Define the common ratio in a geometric sequence.
The constant amount multiplied by each term to get the next term.
What is the initial value in an exponential function?
The value of the function when x = 0 (the 'a' in $f(x) = ab^x$).