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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.

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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 $f(x) = ab^x$ ).

How do you solve an exponential equation?

Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.

How do you solve a logarithmic equation?

Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.

How do you find the inverse of a function?

Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$ .

How do you evaluate a composite function?

Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.

How do you determine if two functions are inverses?

Show that $f(g(x)) = x$ and $g(f(x)) = x$ .

How do you graph an exponential function?

Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.

How do you graph a logarithmic function?

Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.

How do you use properties of logarithms to simplify expressions?

Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.

How to determine the equation of an exponential function from data points?

Start with $f(x) = ab^x$ . 2. Use two data points to create a system of equations. 3. Solve for a and b.

How do you solve for the half-life in an exponential decay problem?

Set up the equation $0.5 = b^t$ , where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.

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?

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

All Flashcards

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 $f(x) = ab^x$ ).

How do you solve an exponential equation?

Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.

How do you solve a logarithmic equation?

Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.

How do you find the inverse of a function?

Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$ .

How do you evaluate a composite function?

Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.

How do you determine if two functions are inverses?

Show that $f(g(x)) = x$ and $g(f(x)) = x$ .

How do you graph an exponential function?

Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.

How do you graph a logarithmic function?

Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.

How do you use properties of logarithms to simplify expressions?

Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.

How to determine the equation of an exponential function from data points?

Start with $f(x) = ab^x$ . 2. Use two data points to create a system of equations. 3. Solve for a and b.

How do you solve for the half-life in an exponential decay problem?

Set up the equation $0.5 = b^t$ , where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.

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?

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$