Podcast
Questions and Answers
What is the correct relationship between the graph of an exponential function and its inverse?
What is the correct relationship between the graph of an exponential function and its inverse?
Which property must a function satisfy to have a unique inverse?
Which property must a function satisfy to have a unique inverse?
What does the expression $y = ext{log}_b(x)$ represent?
What does the expression $y = ext{log}_b(x)$ represent?
How can you solve an exponential equation like $2^x = 8$?
How can you solve an exponential equation like $2^x = 8$?
Signup and view all the answers
What is the domain of the logarithmic function $y = ext{log}_b(x)$?
What is the domain of the logarithmic function $y = ext{log}_b(x)$?
Signup and view all the answers
What occurs when you compose a function and its inverse?
What occurs when you compose a function and its inverse?
Signup and view all the answers
Which of the following statements about logarithmic and exponential functions is true?
Which of the following statements about logarithmic and exponential functions is true?
Signup and view all the answers
If the base of a logarithmic function is negative, what can be concluded?
If the base of a logarithmic function is negative, what can be concluded?
Signup and view all the answers
Study Notes
Inverse of Exponential Functions
- Exponential functions have a unique inverse, known as a logarithmic function.
- The inverse relationship means that if y = bx, then x = logb(y), where b is the base.
- The domain of an exponential function becomes the range of its inverse, and the range of the exponential function becomes the domain of its inverse.
Graphing Inverse Functions
- The graph of an inverse function is a reflection of the original function across the line y = x.
- To graph the inverse of a function, swap the x and y coordinates of points on the original graph.
- The points on the inverse function are essentially the same as those on the original function, but mirrored across the line y = x.
- A function and its inverse are symmetric about the line y = x.
Solving Exponential Equations
- Exponential equations can often be solved using logarithms.
- The key is to isolate the exponential term and then apply the logarithm with the same base as the exponential expression to both sides of the equation to solve for the unknown variable.
- Applying the logarithm of a specific base (e.g., logarithms base 10 (common logarithm) or logarithms base e (natural logarithm)).
- Techniques include applying the properties of logarithms and algebraic manipulation
Logarithmic Functions
- Logarithmic functions are the inverses of exponential functions.
- Logarithmic functions are defined as y = logb(x), where b is the base and x is the argument.
- Logarithms represent the exponent to which the base must be raised to obtain a given number.
- The logarithmic function y = logb(x) is defined only for x > 0 and b > 0, b ≠ 1.
- Common logarithms are base-10 logarithms (log10). Natural logarithms are base-e logarithms (ln = loge).
Properties of Inverse Functions
- The composition of a function and its inverse results in the identity function. f(f-1(x)) = x and f-1(f(x)) = x.
- An inverse function, denoted by f-1(x), reverses the effect of the original function, f(x).
- A function must be one-to-one (injective) to have an inverse function. This means each element in the range corresponds to exactly one element in the domain.
- The domain of the inverse function is the range of the original function; and similarly the range of the inverse function is the domain of the original function.
- If a function is not one-to-one, it does not have a unique inverse, rather it has an inverse relation.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of inverse exponential functions and their properties. This quiz covers the relationship between exponential and logarithmic functions, graphing techniques, and solving exponential equations using logarithms. Challenge yourself to apply these concepts effectively!
