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

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. To determine if two functions are inverses, we check if f(g(x)) = x and g(f(x)) = x for all x in their respective domains.

Logarithmic functions are the inverses of exponential functions. The function f(x) = log₂(x + 1) represents the logarithm base 2 of (x + 1), which means it answers the question: 'To what power must 2 be raised to obtain (x + 1)?' Understanding the properties of logarithms, such as the change of base and the relationship between logs and exponents, is crucial for analyzing inverse relationships.

Exponential functions are of the form g(x) = a^x, where a is a positive constant. In this case, g(x) = 2^(x - 1) represents an exponential function that shifts the graph of 2^x to the right by 1 unit. Recognizing how exponential functions behave, including their growth rates and transformations, is essential for verifying if two functions are inverses.