Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 107

Chapter 5, Problem 107

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2^x-1

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Recall that two functions $ f $ and $ g $ are inverses if and only if $ f(g(x)) = x $ and $ g(f(x)) = x $ for all $ x $ in the domains of the compositions.

Start by finding the composition $ f(g(x)) $. Substitute $ g(x) = 2^{x} - 1 $ into $ f(x) = \log_{2}(x + 1) $, so $ f(g(x)) = \log_{2}((2^{x} - 1) + 1) $.

Simplify the expression inside the logarithm: $ (2^{x} - 1) + 1 = 2^{x} $, so $ f(g(x)) = \log_{2}(2^{x}) $.

Use the logarithm property $ \log_{b}(b^{k}) = k $ to simplify $ \log_{2}(2^{x}) = x $. This shows that $ f(g(x)) = x $.

Next, find the composition $ g(f(x)) $. Substitute $ f(x) = \log_{2}(x + 1) $ into $ g(x) = 2^{x} - 1 $, so $ g(f(x)) = 2^{\log_{2}(x + 1)} - 1 $. Using the property $ b^{\log_{b}(y)} = y $, simplify to $ (x + 1) - 1 = x $. This shows that $ g(f(x)) = x $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

Inverse functions undo each other's operations. If f and g are inverses, then f(g(x)) = x and g(f(x)) = x for all x in their domains. This means applying one function after the other returns the original input.

Properties of Logarithms and Exponents

Logarithms and exponents are inverse operations. Specifically, log base 2 and 2 raised to a power undo each other, such that log₂(2^x) = x and 2^(log₂ x) = x. Understanding this relationship helps verify if two functions are inverses.

Function Composition

Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). To check if two functions are inverses, compose them in both orders and verify if the result simplifies to x.

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Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2x-1

Verified video answer for a similar problem:

Key Concepts

Inverse Functions

Properties of Logarithms and Exponents

Function Composition

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2^x-1