Textbook Question
Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 5x + 1
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 112
Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)x
Verified step by step guidance1
Start with the given function: \(f(x) = \left(\frac{1}{3}\right)^x\).
To find the inverse function, first replace \(f(x)\) with \(y\): \(y = \left(\frac{1}{3}\right)^x\).
Swap the variables \(x\) and \(y\) to get the inverse relation: \(x = \left(\frac{1}{3}\right)^y\).
To solve for \(y\), take the logarithm of both sides. You can use the natural logarithm or logarithm base 10: \(\log\left(x\right) = \log\left(\left(\frac{1}{3}\right)^y\right)\).
Use the logarithm power rule to bring down the exponent: \(\log(x) = y \cdot \log\left(\frac{1}{3}\right)\). Then solve for \(y\): \(y = \frac{\log(x)}{\log\left(\frac{1}{3}\right)}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A one-to-one function is a function where each output corresponds to exactly one input, ensuring it has an inverse. This means no two different inputs produce the same output, which is essential for the inverse function to exist.
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Decomposition of Functions
Inverse Functions
The inverse of a function reverses the roles of inputs and outputs, effectively 'undoing' the original function. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
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4:30Graphing Logarithmic Functions
Exponential and Logarithmic Functions
Exponential functions like f(x) = (1/3)^x have inverses that are logarithmic functions. To find the inverse, you rewrite the equation in logarithmic form, using the base of the exponential to solve for x.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2x-1
Textbook Question
Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 3x
Textbook Question
Find ƒ-1(x), and give the domain and range. ƒ(x) = ex-5
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Textbook Question
Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 4x+2
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Textbook Question
Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4x + 3
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