Answered: Inverse Functions 24 (a) Find the…

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Calculus

Inverse Functions 24 (a) Find the inverse of the function f(x) 1+ 2 (b) What is the domain of the inverse function?

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter4: Exponential And Logarithmic Functions

Section4.3: Logarithmic Functions

Problem 95E: Inverse Functions Find the inverse of the function f(x)=2x1+2x What is the domain of the inverse...

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Question

Inverse Functions
24
(a) Find the inverse of the function f(x)
1+ 2
(b) What is the domain of the inverse function?

Expert Solution

Step 1

Here given function as

$f (x) = \frac{2^{x}}{1 + 2^{x}}$

Need to find the inverse function and the domain of the inverse function.

Step 2

(a)

$f (x) = \frac{2^{x}}{1 + 2^{x}} = y (say)$

Then

$\Rightarrow 2^{x} = (1 + 2^{x}) y \Rightarrow 2^{x} = y + 2^{x} y \Rightarrow 2^{x} (1 - y) = y \Rightarrow 2^{x} = \frac{y}{1 - y} \Rightarrow \log_{2} (2^{x}) = \log_{2} (\frac{y}{1 - y}) [By taking \log on both sides] \Rightarrow x = \log_{2} (\frac{y}{1 - y}) [∵ \log_{a} (a^{x}) = x]$

replace $x$ by $f^{- 1} (x)$ and $y$ by $x$ , then

$f^{- 1} (x) = \log_{2} (\frac{x}{1 - x})$

Step by step

Solved in 3 steps

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