Reciprocal Function Transformation - Algebra-2

1. Fundamental Concepts

Definition: A reciprocal function is a function of the form $$f(x) = \frac{1}{x}$$, where $$x \neq 0$$.
Domain and Range: The domain of $$f(x) = \frac{1}{x}$$ is all real numbers except $$x = 0$$. The range is also all real numbers except $$y = 0$$.
Asymptotes: The x-axis ($$y = 0$$) and y-axis ($$x = 0$$) are asymptotes for the graph of $$f(x) = \frac{1}{x}$$.

2. Key Concepts

Basic Rule: $$f(x) = \frac{1}{x}$$

Transformation Rules: For a function $$g(x) = \frac{a}{x - h} + k$$, the graph shifts horizontally by $$h$$ units and vertically by $$k$$ units, and scales by a factor of $$a$$.

Application: Reciprocal functions are used in physics to model inverse relationships, such as the relationship between distance and force in Hooke's Law.

3. Examples

Example 1 (Basic)

Problem: Graph the function $$f(x) = \frac{1}{x}$$.

Step-by-Step Solution:

Identify the vertical and horizontal asymptotes: $$x = 0$$ and $$y = 0$$.
Plot key points: For example, at $$x = 1$$, $$f(1) = 1$$; at $$x = -1$$, $$f(-1) = -1$$.
Draw the graph using the asymptotes and key points.

Validation: Check that the graph approaches the asymptotes but never touches them.

Example 2 (Intermediate)

Problem: Graph the function $$g(x) = \frac{2}{x - 3} + 1$$.

Step-by-Step Solution:

Identify the vertical and horizontal asymptotes: $$x = 3$$ and $$y = 1$$.
Plot key points: For example, at $$x = 4$$, $$g(4) = \frac{2}{4 - 3} + 1 = 3$$; at $$x = 2$$, $$g(2) = \frac{2}{2 - 3} + 1 = -1$$.
Draw the graph using the asymptotes and key points.

Validation: Check that the graph approaches the asymptotes but never touches them.

4. Problem-Solving Techniques

Graphical Analysis: Use graphs to visualize transformations and understand the behavior of reciprocal functions.
Algebraic Manipulation: Apply algebraic rules to simplify expressions involving reciprocal functions.
Contextual Understanding: Relate reciprocal functions to real-world scenarios to enhance understanding.