Inverse Relations and Functions - Algebra-2

1. Fundamental Concepts

Definition: Inverse relations and functions are pairs of functions where one function reverses the effect of the other. If $f$ is a function, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
One-to-One Functions: A function has an inverse if it is one-to-one (each output corresponds to exactly one input).
Graphical Interpretation: The graph of an inverse function is a reflection of the original function over the line $y = x$.

Basic Rule: $$f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x$$

Degree Preservation: The highest degree in the result matches input

Application: Used to solve equations and model real-world scenarios such as temperature conversion

Problem: Find the inverse of the function $f(x) = 2x + 3$.

Validation: Substitute $x = 5$: Original: $f(5) = 2(5) + 3 = 13$; Simplified: $f^{-1}(13) = \frac{13 - 3}{2} = 5$ ✓

Problem: Determine if the function $g(x) = x^3 + 2$ is invertible and find its inverse.

Validation: Substitute $x = 3$: Original: $g(3) = 3^3 + 2 = 29$; Simplified: $g^{-1}(29) = \sqrt[3]{29 - 2} = 3$ ✓

Visual Strategy: Graph both the function and its inverse on the same coordinate system to visually confirm they are reflections over $y = x$.
Error-Proofing: Always verify the solution by substituting a value into both the function and its inverse.
Concept Reinforcement: Practice with various types of functions (linear, quadratic, cubic) to reinforce understanding of inverses.