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

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 \( g(x) = 4^x + 3 \) into \( f(x) = \log_4 (x + 3) \), so \( f(g(x)) = \log_4 ((4^x + 3) + 3) \).
3. Simplify the expression inside the logarithm: \( (4^x + 3) + 3 = 4^x + 6 \). So, \( f(g(x)) = \log_4 (4^x + 6) \).
4. Next, find the composition \( g(f(x)) \). Substitute \( f(x) = \log_4 (x + 3) \) into \( g(x) = 4^x + 3 \), so \( g(f(x)) = 4^{\log_4 (x + 3)} + 3 \).
5. Simplify \( 4^{\log_4 (x + 3)} \) using the property that \( a^{\log_a b} = b \), so \( g(f(x)) = (x + 3) + 3 = x + 6 \). Compare both compositions to \( x \) to determine if \( f \) and \( g \) are inverses.

Key Concepts

Inverse Functions

Inverse functions reverse each other's operations, meaning if f and g are inverses, then f(g(x)) = x and g(f(x)) = x for all x in their domains. Understanding this relationship is essential to verify if two functions are inverses.

Logarithmic and Exponential Functions

Logarithmic functions are the inverses of exponential functions with the same base. For example, log base 4 and 4 raised to a power undo each other, which is key to analyzing the given functions f(x) = log₄(x+3) and g(x) = 4^x + 3.

Function Composition

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