Answered: Show that f (x) = x + x - 1 is…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Show that f (x) = x + x ^{- 1} is one-to-one on [1, oo), and find the corresponding inverse f^{- 1}. What is the domain of f^{- 1}?

Expert Solution

Step 1

Given:

$f (x) = x + x^{- 1}$

Claim:

$f (x)$ is one-to -one function.

Proof:

A function is said to be one-to-one function if for every x-value in the domain, there exists a unique y-value in the Co-domain of the given function.

Step 2

So,

This function can be written as :

$f (x) = x + \frac{1}{x}$

To prove the function as one-to-one:

Put

$f (x) = f (y)$

$x + \frac{1}{x} = y + \frac{1}{y} x - y = \frac{1}{y} - \frac{1}{x} x - y = \frac{x - y}{x y} x y = 1 y = \frac{1}{x} y = x^{- 1}$

Hence,

For every element the function has an unique value, so the given function is one-one on [1, $\infty$ ).

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