Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4^x + 3

Key Concepts

Inverse Functions

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 hold true for all x in their respective domains.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The function f(x) = log₄(x + 3) represents the logarithm base 4 of (x + 3), which answers the question: 'To what power must 4 be raised to obtain (x + 3)?' 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

Exponential functions are of the form g(x) = a^x, where a is a positive constant. In this case, g(x) = 4^x + 3 represents an exponential function shifted vertically by 3 units. Recognizing how exponential functions behave, including their growth rates and transformations, is essential for verifying if two functions are inverses, particularly when combined with logarithmic functions.