Rational Functions and Their Graphs

Definition of Rational Functions

Rational functions are defined as quotients of polynomial functions.
They can be expressed in the form $f(x) = \frac{p(x)}{q(x)}$ .
In this form, $p(x)$ and $q(x)$ are polynomial functions.
A crucial condition is that the denominator $q(x)$ cannot be equal to zero, i.e., $q(x) \neq 0$ .

Domain of Rational Functions

The domain of a rational function consists of all real numbers for which the denominator is not zero.
To find the domain, set the denominator equal to zero and solve for $x$ . These $x$ -values are excluded from the domain.
Example 1: Find the domain of each rational function:
1. $f(x) = \frac{1}{x-3}$
  - Denominator: $x-3 = 0 \implies x=3$
  - Domain: All real numbers except $x=3$ , or $(-\infty, 3) \cup (3, \infty)$ .
2. $g(x) = \frac{(x+3)(x-3)}{x-3}$
  - Denominator: $x-3 = 0 \implies x=3$
  - Domain: All real numbers except $x=3$ .
  - Note: While $(x-3)$ is a common factor, the original function is undefined at $x=3$ .
3. $h(x) = \frac{x}{x^2-9}$
  - Denominator: $x^2-9 = 0 \implies (x-3)(x+3) = 0 \implies x=3 \text{ or } x=-3$
  - Domain: All real numbers except $x=3$ and $x=-3$ , or $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$ .
4. $k(x) = \frac{x+3}{x^2+9}$
  - Denominator: $x^2+9 = 0 \implies x^2 = -9$
  - There are no real solutions for $x$ since $x^2$ cannot be negative.
  - Domain: All real numbers, or $(-\infty, \infty)$ .

Reciprocal Function

The simplest rational function is the reciprocal function.
It is defined by $f(x) = \frac{1}{x}$ .

Graphing Rational Functions

Example 2: Graph the function $f(x) = \frac{1}{x^2}$
- This function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .
- The graph is symmetric with respect to the y-axis.
Example 3: Graph the function $f(x) = \frac{1}{(x-2)^2} + 1$
- This is a transformation of $f(x) = \frac{1}{x^2}$ . It is shifted 2 units to the right and 1 unit up.
- Vertical asymptote at $x=2$ . Horizontal asymptote at $y=1$ .

Vertical Asymptotes of Rational Functions

The line $x = a$ is a vertical asymptote of the graph of a function $f$ if $f(x)$ increases or decreases without bound as $x$ approaches $a$ (from either side).
Condition for Rational Functions:
- For a rational function $f(x) = \frac{p(x)}{q(x)}$ (where $p(x)$ and $q(x)$ have no common factors other than constants):
  - If $(x-a)$ is a factor of $q(x)$ but not a factor of $p(x)$ , then $x=a$ is a vertical asymptote of the graph of $f$ .
- Holes in the Graph:
  - If $(x-a)$ is a common factor of both $p(x)$ and $q(x)$ , and if there is a reduced form of the function's equation where $x=a$ does not make the denominator zero, then $x=a$ corresponds to a hole in the graph, not a vertical asymptote.
Example 4: Find the vertical asymptotes, if any, and the values of $x$ corresponding to holes, if any, of the graph of each rational function:
1. $f(x) = \frac{x}{x^2-9} = \frac{x}{(x-3)(x+3)}$
  - Denominator factors are $(x-3)$ and $(x+3)$ . Neither is a factor of the numerator.
  - Vertical asymptotes: $x=3$ and $x=-3$ .
  - No holes.
2. $g(x) = \frac{x+3}{x^2-9} = \frac{x+3}{(x-3)(x+3)}$
  - Common factor: $(x+3)$ . Reduced form: $g(x) = \frac{1}{x-3}$ for $x \neq -3$ .
  - Vertical asymptote: $x=3$ (from $(x-3)$ ).
  - Hole: At $x=-3$ .
3. $h(x) = \frac{x+3}{x^2+9}$
  - Denominator: $x^2+9$ has no real roots.
  - No vertical asymptotes.
  - No holes.
4. $k(x) = \frac{x^2-9}{x+3} = \frac{(x-3)(x+3)}{x+3}$
  - Common factor: $(x+3)$ . Reduced form: $k(x) = x-3$ for $x \neq -3$ .
  - No vertical asymptotes (the denominator of the reduced form is 1, which is never zero).
  - Hole: At $x=-3$ .

Horizontal Asymptotes of Rational Functions

The line $y = b$ is a horizontal asymptote of the graph of a function $f$ if $f(x)$ approaches $b$ as $x$ increases or decreases without bound ( $x \to \infty$ or $x \to -\infty$ ).
Rules for finding Horizontal Asymptotes for $f(x) = \frac{an x^n + a{n-1} x^{n-1} + \cdots + a1 x + a0}{bm x^m + b{m-1} x^{m-1} + \cdots + b1 x + b0}$ (where $an \neq 0$ and $bm \neq 0$ ):
1. If the degree of the numerator (n) is less than the degree of the denominator (m) (n < m):
 - The x-axis, or $y=0$ , is the horizontal asymptote of the graph of $f$ .
2. If the degree of the numerator (n) is equal to the degree of the denominator (m) ( $n = m$ ):
 - The line $y = \frac{an}{bm}$ (ratio of the leading coefficients) is the horizontal asymptote of the graph of $f$ .
3. If the degree of the numerator (n) is greater than the degree of the denominator (m) (n > m):
 - The graph of $f$ has no horizontal asymptote.
Example 5: Find the horizontal asymptotes, if any, of the graph of each rational function:
1. $f(x) = \frac{4x}{2x^2+1}$
 - Numerator degree $n=1$ . Denominator degree $m=2$ . (n < m)
 - Horizontal asymptote: $y=0$ .
2. $g(x) = \frac{4x^2}{2x^2+1}$
 - Numerator degree $n=2$ . Denominator degree $m=2$ . ( $n = m$ )
 - Horizontal asymptote: $y = \frac{4}{2} = 2$ .
3. $h(x) = \frac{4x^3}{2x^2+1}$
 - Numerator degree $n=3$ . Denominator degree $m=2$ . (n > m)
 - No horizontal asymptote.

Graphing Rational Functions (Advanced Examples)

Example 6: Graph $f(x) = \frac{2x-1}{x-1}$
- Vertical asymptote at $x=1$ .
- Horizontal asymptote at $y=2$ (degree of numerator = degree of denominator, ratio of leading coefficients is $2/1=2$ ).
Example 7: Graph $f(x) = \frac{3x^2}{x^2-4}$
- Vertical asymptotes at $x=2$ and $x=-2$ (from $x^2-4=0$ ).
- Horizontal asymptote at $y=3$ (degree of numerator = degree of denominator, ratio of leading coefficients is $3/1=3$ ).

Slant (Oblique) Asymptotes

The graph of a rational function has a slant (or oblique) asymptote if the degree of the numerator is exactly one more than the degree of the denominator.
To find the equation of a slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient (excluding the remainder) is the equation of the slant asymptote.
Example 8: Find the slant asymptote of the graph of the rational function $f(x) = \frac{x^2+1}{x-1}$
- Degree of numerator ( $2$ ) is one more than the degree of the denominator ( $1$ ).
- Using polynomial long division ( $(x^2+1) \div (x-1)$ ):
  x + 1 x-1 | x^2 + 0x + 1 -(x^2 - x) ---------- x + 1 -(x - 1) --------- 2
- The quotient is $x+1$ with a remainder of $2$ . So, $f(x) = x+1 + \frac{2}{x-1}$ .
- As $x \to \infty$ or $x \to -\infty$ , the term $\frac{2}{x-1}$ approaches $0$ .
- The slant asymptote is $y = x+1$ .

Writing the Equation of a Rational Function from Given Properties

Example 9: If $f$ has a vertical asymptote given by $x=7$ , a horizontal asymptote $y=0$ , y-intercept at $-2$ , and no x-intercept, then write the equation of a rational function $f(x) = \frac{p(x)}{q(x)}$ having the indicated properties, in which the degrees of $p$ and $q$ are as small as possible.
- Vertical Asymptote $x=7$ : This implies $(x-7)$ is a factor of the denominator $q(x)$ .
- Horizontal Asymptote $y=0$ : This implies the degree of the numerator (n) must be less than the degree of the denominator (m) (n < m).
- No x-intercept: This means $p(x)$ never equals zero. If $p(x)$ is a constant, it won't have x-intercepts.
- y-intercept at $-2$ : This means $f(0) = -2$ .
- Let's try the simplest form: $p(x)$ is a constant and $q(x)$ has $(x-7)$ as a factor.
 - Let $f(x) = \frac{C}{x-7}$ . Here, $n=0$ and $m=1$ , so n < m satisfies $y=0$ horizontal asymptote.
 - No x-intercept is satisfied since a constant numerator never equals zero.
 - Use the y-intercept: $f(0) = \frac{C}{0-7} = -2$
 - $\frac{C}{-7} = -2 \implies C = 14$
- Therefore, the function is $f(x) = \frac{14}{x-7}$ .

Application of Rational Functions: Population Modeling

Example 10: The following rational function, in hundreds, models the population of a certain species of animal, where $x$ is in days. What number does the population approach in the long run? $p(x) = \frac{10x^3 + 200}{2x^3 + 1.6}$