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

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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.

Start by finding the composition $ f(g(x)) $. Substitute $ g(x) = \log_5 x $ into $ f $: \[ f(g(x)) = f(\log_5 x) = 5^{\log_5 x} \].

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 $.

Next, find the composition $ g(f(x)) $. Substitute $ f(x) = 5^x $ into $ g $: \[ g(f(x)) = g(5^x) = \log_5 (5^x) \].

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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