Inverse Function - Algebra-1

1. Fundamental Concepts

Definition: An inverse function, denoted as $f^{-1}$, is a function that reverses the operation of another function $f$. If $f(a) = b$, then $f^{-1}(b) = a$.
One-to-One Function: A function has an inverse if and only 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 $y = x$.

2. Key Concepts

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

Degree Preservation: The domain of $f^{-1}$ is the range of $f$ and vice versa.

Application: Inverse functions are used in various fields such as cryptography, physics, and engineering to solve equations and model relationships.

3. Examples

Example 1 (Basic)

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

Step-by-Step Solution:

Let $y = f(x)$: $y = 2x + 3$
Solve for $x$: $y - 3 = 2x \Rightarrow x = \frac{y - 3}{2}$
Interchange $x$ and $y$: $y = \frac{x - 3}{2}$
The inverse function is $f^{-1}(x) = \frac{x - 3}{2}$

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

Example 2 (Intermediate)

Problem: Find the inverse of the function $g(x) = \sqrt{x - 4}$.

Step-by-Step Solution:

Let $y = g(x)$: $y = \sqrt{x - 4}$
Square both sides: $y^2 = x - 4$
Solve for $x$: $x = y^2 + 4$
Interchange $x$ and $y$: $y = x^2 + 4$
The inverse function is $g^{-1}(x) = x^2 + 4$

Validation: Substitute $x = 6$: Original: $g(6) = \sqrt{6 - 4} = 2$; Simplified: $g^{-1}(2) = 2^2 + 4 = 8$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use graphs to visualize the relationship between a function and its inverse.
Error-Proofing: Always check the domain and range when finding the inverse to ensure it is valid.
Concept Reinforcement: Practice with different types of functions (linear, quadratic, etc.) to reinforce understanding.