Graph Rational Functions

1. Fundamental Concepts
Definition: A rational function is a function of the form
$$f(x) = \frac{P(x)}{Q(x)},$$
where $P(x)$ and $Q(x)$ are polynomials, and $Q(x) \neq 0$.
Domain: All real numbers except where the denominator equals zero.
Range: Depends on the function, but is often restricted by asymptotes or undefined values.
Key Behavior: Rational functions often have asymptotes that guide their shape and can show intercepts, symmetry, and discontinuities (holes).

2. Key Concepts
Vertical Asymptotes: Occur where the denominator $Q(x) = 0$.
Horizontal Asymptotes: Determined by comparing the degrees of numerator and denominator:

If $\deg P < \deg Q$, then $y = 0$.
If $\deg P = \deg Q$, then $y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$.
If $\deg P > \deg Q$, there is no horizontal asymptote (but possibly an oblique/slant one).
Intercepts:
x-intercepts: Solve $P(x) = 0$ (numerator = 0, denominator ≠ 0).
y-intercept: Evaluate $f(0)$, if defined.
Holes: If a factor cancels from numerator and denominator, the graph has a hole at that x-value.

3. Examples
Example 1 (Basic)
Problem: Graph
$$f(x) = \frac{1}{x}.$$

Step-by-Step Solution:
Domain: $x \neq 0$.
Vertical Asymptote: $x = 0$.
Horizontal Asymptote: $y = 0$.
Intercepts: None.
Behavior: Decreases in Quadrants I and III.

Example 2 (Intermediate)
Problem: Graph
$$f(x) = \frac{x^2 - 9}{x - 3}.$$
Step-by-Step Solution:
Factor numerator: $f(x) = \frac{(x - 3)(x + 3)}{x - 3}$.
Simplify: $f(x) = x + 3$, except at $x = 3$.
Domain: All real numbers except $x = 3$.
Hole: At $x = 3$. The y-coordinate is $3 + 3 = 6$, so hole at $(3, 6)$.
Graph: Looks like the line $y = x + 3$ but with a hole at $(3,6)$.

Example 3 (Advanced)
Problem: Graph
$$f(x) = \frac{2x^2 + 3}{x^2 - 1}.$$
Step-by-Step Solution:
Domain: $x \neq \pm 1$.
Vertical Asymptotes: $x = 1, \, -1$.
Degrees: numerator degree = 2, denominator degree = 2 → horizontal asymptote at $y = \frac{2}{1} = 2$.
x-intercepts: Solve $2x^2 + 3 = 0$. No real solutions → no x-intercept.
y-intercept: $f(0) = \frac{3}{-1} = -3$.
Behavior: Graph approaches vertical asymptotes at $x = \pm 1$ and horizontal asymptote at $y = 2$.

4. Problem-Solving Techniques
Find Asymptotes First: They provide the “skeleton” of the graph.
Check for Holes: Simplify expressions to see if any factor cancels.
Plot Intercepts: Locate where the graph crosses axes.
Test Intervals: Choose test points between asymptotes to check function behavior.
Use End Behavior: Horizontal (or slant) asymptotes describe how the function behaves as $x \to \pm\infty$.