Textbook Question
Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.3.42
Find the inverse of each function (on the given interval, if specified).
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Verified step by step guidance1
To find the inverse of the function \( f(x) = \frac{x}{x-2} \), start by replacing \( f(x) \) with \( y \). So, we have \( y = \frac{x}{x-2} \).
Next, solve for \( x \) in terms of \( y \). Begin by multiplying both sides by \( x-2 \) to eliminate the fraction: \( y(x-2) = x \).
Distribute \( y \) on the left side: \( yx - 2y = x \).
Rearrange the equation to isolate terms involving \( x \) on one side: \( yx - x = 2y \).
Factor out \( x \) from the left side: \( x(y - 1) = 2y \). Finally, solve for \( x \) by dividing both sides by \( y - 1 \): \( x = \frac{2y}{y - 1} \). This expression represents \( f^{-1}(x) \) when you replace \( y \) with \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f^{-1} takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.
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Finding Inverses Algebraically
To find the inverse of a function algebraically, you typically start by replacing f(x) with y. Then, you solve the equation for x in terms of y. Finally, you swap x and y to express the inverse function. This process often involves algebraic manipulation, such as isolating variables and simplifying expressions.
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Domain and Range Considerations
When determining the inverse of a function, it is crucial to consider the domain and range of both the original function and its inverse. The domain of the original function becomes the range of the inverse, and vice versa. This is particularly important for functions that are not defined for all real numbers, as it affects the validity of the inverse function.
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