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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.3.42
Chapter 1, Problem 1.3.42

Find the inverse f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).
f(x)=xx−2f\(\left\)(x\(\right\))=\(\frac{x}{x-2}\), for x>2x\(\gt{2}\)

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1
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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Related Practice
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Compute ƒ(2) and ƒ(y²).

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Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

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