Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = 5^x, g(x) = log￬5 x

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 \( g \) and \( f \), respectively.
2. Start by finding the composition \( f(g(x)) \). Substitute \( g(x) = \log_5 x \) into \( f \): \[ f(g(x)) = f(\log_5 x) = 5^{\log_5 x} \].
3. Use the property of logarithms and exponents that \( a^{\log_a x} = x \) to simplify \( 5^{\log_5 x} \) to \( x \). This shows that \( f(g(x)) = x \).
4. Next, find the composition \( g(f(x)) \). Substitute \( f(x) = 5^x \) into \( g \): \[ g(f(x)) = g(5^x) = \log_5 (5^x) \].
5. Use the logarithm property \( \log_a (a^x) = x \) to simplify \( \log_5 (5^x) \) to \( x \). This shows that \( g(f(x)) = x \). Since both compositions simplify to \( x \), \( f \) and \( g \) are inverses.

Key Concepts

Inverse Functions

Inverse functions undo each other's operations. If f and g are inverses, then applying f followed by g (or vice versa) returns the original input, i.e., g(f(x)) = x and f(g(x)) = x for all x in the domains. This property is essential to verify if two functions are inverses.

Exponential Functions

An exponential function has the form f(x) = a^x, where a is a positive constant not equal to 1. It grows rapidly and is one-to-one, meaning it passes the horizontal line test, which allows it to have an inverse function. Understanding its behavior helps in analyzing its inverse.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is defined as g(x) = log_a(x), where a is the base. It satisfies the property that log_a(a^x) = x and a^(log_a x) = x. Recognizing this relationship is key to determining if two functions are inverses.