One-to-One Function - Algebra-2

1. Fundamental Concepts

Definition: A one-to-one function is a function where each element in the domain maps to exactly one element in the range, and no two different elements in the domain map to the same element in the range.
Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.
Inverse Function: If $f$ is a one-to-one function, then its inverse function $f^{-1}$ exists such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

2. Key Concepts

Basic Rule: $$f(a) = f(b) \implies a = b$$

Determining One-to-One: Use the Horizontal Line Test: Every horizontal line intersects the graph of $f$ at most once.

Application: One-to-one functions are crucial for defining and using inverse functions in various applications, including solving equations and modeling real-world phenomena.

3. Examples

Example 1 (Basic)

Problem: Determine if the function $f(x) = 2x + 3$ is one-to-one.

Step-by-Step Solution:

Assume $f(a) = f(b)$: $2a + 3 = 2b + 3$
Solve for $a$ and $b$: $2a = 2b \implies a = b$

Validation: Since $a = b$ whenever $f(a) = f(b)$, the function is one-to-one.

Example 2 (Intermediate)

Problem: Find the inverse of the function $g(x) = 4x - 5$.

Step-by-Step Solution:

Set $y = g(x)$: $y = 4x - 5$
Solve for $x$: $y + 5 = 4x \implies x = \frac{y + 5}{4}$
Interchange $x$ and $y$: $y = \frac{x + 5}{4}$

Validation: The inverse function is $g^{-1}(x) = \frac{x + 5}{4}$. Check: $g(g^{-1}(x)) = 4\left(\frac{x + 5}{4}\right) - 5 = x$.

4. Problem-Solving Techniques

Graphical Method: Use the Horizontal Line Test to visually determine if a function is one-to-one.
Algebraic Verification: Assume $f(a) = f(b)$ and solve to show $a = b$.
Function Composition: Verify the inverse by showing $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.