Reciprocal Function Graph - 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 graph of $$f(x) = \frac{1}{x}$$ has two asymptotes: the vertical asymptote at $$x = 0$$ and the horizontal asymptote at $$y = 0$$.

2. Key Concepts

Behavior Near Asymptotes: As $$x \rightarrow 0^+$$, $$f(x) \rightarrow +\infty$$; as $$x \rightarrow 0^-$$, $$f(x) \rightarrow -\infty$$.

Symmetry: The graph of $$f(x) = \frac{1}{x}$$ is symmetric with respect to the origin.

Application: Reciprocal functions are used in various applications such as modeling electrical circuits and describing inverse relationships in physics.

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, when $$x = 1$$, $$f(x) = 1$$; when $$x = -1$$, $$f(x) = -1$$.
Draw the graph by connecting these points smoothly, approaching the asymptotes but never touching them.

Validation: Check that the graph approaches the asymptotes correctly and maintains symmetry about the origin.

Example 2 (Intermediate)

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

Step-by-Step Solution:

Identify the vertical asymptote: $$x = 2$$.
Identify the horizontal asymptote: $$y = 0$$.
Plot key points: For example, when $$x = 3$$, $$g(x) = 1$$; when $$x = 1$$, $$g(x) = -1$$.
Draw the graph by connecting these points smoothly, approaching the asymptotes but never touching them.

Validation: Check that the graph approaches the new vertical asymptote at $$x = 2$$ and maintains the horizontal asymptote at $$y = 0$$.

4. Problem-Solving Techniques

Graphing Strategy: Always start by identifying the asymptotes and plotting key points around them.
Transformation Analysis: Understand how transformations affect the graph of the reciprocal function, such as shifts and stretches.
Contextual Understanding: Relate the graph to real-world applications to deepen understanding.