Finding Inverses of Exponential Logarithmic Functions - Precalculus for Teens!
Miacademy Learning Channel・2 minutes read
Exponential and logarithmic functions are inverse functions when they share the same base, and their relationship is confirmed when compositions of their respective functions yield the input value. The text also demonstrates how to find and analyze the inverses of specific functions, like \( C(x) \) and \( D(x) \), and explains the derivation of sound intensity as a function of sound level in decibels.
Insights
- Exponential and logarithmic functions are fundamentally linked as inverse functions, meaning their graphs mirror each other across the line y = x, which is illustrated by the equations \( b^{\log_b(x)} = x \) and \( \log_b(b^x) = x \); this relationship is crucial for understanding how to manipulate these functions effectively in mathematical contexts.
- To determine if two functions are inverses, one must confirm that their compositions yield the identity function, as demonstrated with \( A(x) = \log_{10}(x 2) + 5 \) and \( B(x) = 10^{(x - 5)} \); this principle is essential not only for verifying inverse relationships but also for practical applications in solving equations and transforming functions.
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Recent questions
What are exponential functions used for?
Exponential functions are widely used in various fields such as finance, biology, and physics to model growth processes, decay, and complex systems. In finance, they help in calculating compound interest, where the amount grows exponentially over time. In biology, they can describe population growth, where the number of individuals increases rapidly under ideal conditions. In physics, exponential functions are used to model radioactive decay, where the quantity of a substance decreases exponentially over time. Their unique properties, such as rapid growth or decay, make them essential for understanding and predicting real-world phenomena.
How do I find a function's inverse?
To find a function's inverse, you start by replacing the function notation, typically \( f(x) \), with \( y \). Next, you swap the variables \( x \) and \( y \) to reflect the inverse relationship. After that, you solve the resulting equation for \( y \) to express it in terms of \( x \). This new equation represents the inverse function, denoted as \( f^{-1}(x) \). It’s important to verify that the original function and its inverse satisfy the condition that applying one after the other returns the original input, ensuring they are true inverses.
What is the domain of a function?
The domain of a function refers to the set of all possible input values (or x-values) for which the function is defined. It is crucial to identify the domain because certain values may lead to undefined expressions, such as division by zero or taking the square root of a negative number. For example, in the function \( f(x) = \sqrt{x} \), the domain is limited to non-negative numbers since the square root of a negative number is not defined in the real number system. Understanding the domain helps in accurately graphing the function and predicting its behavior.
What is a logarithmic function?
A logarithmic function is the inverse of an exponential function and is defined as \( y = \log_b(x) \), where \( b \) is the base. It answers the question: "To what exponent must the base \( b \) be raised to produce \( x \)?" Logarithmic functions are essential in various applications, including solving equations involving exponentials, measuring sound intensity in decibels, and analyzing data that spans several orders of magnitude. They have unique properties, such as the ability to transform multiplicative relationships into additive ones, making them valuable in fields like science, engineering, and finance.
Why are inverse functions important?
Inverse functions are important because they allow us to reverse the effects of a function, providing a way to solve equations and understand relationships between variables. For instance, if a function models a process, its inverse can help determine the original input from a given output. This is particularly useful in fields like mathematics, physics, and engineering, where understanding the relationship between variables is crucial. Additionally, inverse functions help in graphing and analyzing functions, as they reveal symmetry and can simplify complex calculations, making them a fundamental concept in mathematics.
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Summary
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Inverse Functions and Their Properties Explained
- Exponential and logarithmic functions are inverse functions of each other when they share the same base, resulting in their graphs being reflections over the line y = x. This relationship is established through the properties: \( b^{\log_b(x)} = x \) and \( \log_b(b^x) = x \).
- To verify if two functions A and B are inverses, one must prove that \( A(B(x)) = x \) and \( B(A(x)) = x \). For example, using \( A(x) = \log_{10}(x - 2) + 5 \) and \( B(x) = 10^{(x - 5)} \), both compositions simplify to x, confirming they are inverses.
- To find the inverse of the function \( C(x) = 5 - 6 \cdot 3^{(x + 2)} \), follow these steps: replace \( C(x) \) with y, swap x and y, and solve for y. The final inverse is \( C^{-1}(x) = \log_3\left(-\frac{x - 5}{6}\right) - 2 \).
- The domain of \( C(x) \) is from negative infinity to 5 (non-inclusive), and its range is from negative infinity to 5 (non-inclusive). The domain of \( C^{-1}(x) \) becomes from negative infinity to 5 (non-inclusive), while its range is from negative infinity to infinity.
- For the logarithmic function \( D(x) = \frac{1}{3} \ln(x - 2) - 7 \), the inverse is found by replacing \( D(x) \) with y, swapping x and y, and isolating y, resulting in \( D^{-1}(x) = e^{3x + 21} + 2 \).
- The domain of \( D(x) \) is from 2 (non-inclusive) to infinity, and its range is from negative infinity to infinity. Consequently, the domain of \( D^{-1}(x) \) is from negative infinity to infinity, while its range is from 2 to infinity.
- To rewrite the equation \( \beta = 10 \log\left(\frac{I}{10^{-12}}\right) \) to show I as a function of beta, divide both sides by 10, convert to exponential form, and multiply by \( 10^{-12} \), yielding \( I = 10^{\frac{\beta}{10}} \cdot 10^{-12} \).
- The process of finding the inverse of the function relating sound intensity I and sound level in decibels β illustrates that while the variables I and β are not swapped, the relationship remains valid, confirming that they are indeed inverse functions.
