Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x+1/x-2, g(x) = 2x+1/x-1

Verified step by step guidance

1. To determine if two functions \( f(x) \) and \( g(x) \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).
2. First, find \( f(g(x)) \). Substitute \( g(x) = \frac{2x+1}{x-1} \) into \( f(x) = \frac{x+1}{x-2} \).
3. Calculate \( f(g(x)) = \frac{\left(\frac{2x+1}{x-1}\right) + 1}{\left(\frac{2x+1}{x-1}\right) - 2} \). Simplify the expression to see if it equals \( x \).
4. Next, find \( g(f(x)) \). Substitute \( f(x) = \frac{x+1}{x-2} \) into \( g(x) = \frac{2x+1}{x-1} \).
5. Calculate \( g(f(x)) = \frac{2\left(\frac{x+1}{x-2}\right) + 1}{\left(\frac{x+1}{x-2}\right) - 1} \). Simplify the expression to see if it equals \( x \).

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. For functions f and g to be inverses, the composition of f(g(x)) and g(f(x)) must equal x for all x in their domains. This means that applying one function followed by the other returns the original input.

Composition of Functions

The composition of functions involves combining two functions to create a new function. If f and g are two functions, their composition is denoted as f(g(x)) or g(f(x)). Evaluating these compositions is essential for verifying if two functions are inverses, as it tests whether they yield the identity function.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. When determining if two functions are inverses, it is crucial to consider their domains and ranges, as they must align appropriately for the functions to be valid inverses of each other.