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

Verified step by step guidance

1. 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.
2. Start by finding the composition \( f(g(x)) \). Substitute \( g(x) = 2^{x} - 1 \) into \( f(x) = \log_{2}(x + 1) \), so \( f(g(x)) = \log_{2}((2^{x} - 1) + 1) \).
3. Simplify the expression inside the logarithm: \( (2^{x} - 1) + 1 = 2^{x} \), so \( f(g(x)) = \log_{2}(2^{x}) \).
4. Use the logarithm property \( \log_{b}(b^{k}) = k \) to simplify \( \log_{2}(2^{x}) = x \). This shows that \( f(g(x)) = x \).
5. 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 \( g(f(x)) = x \).

Key Concepts

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.