Rational Functions

A rational function is any function of the form $$ f(x) = \frac{P(x)}{Q(x)} $$ where $P(x)$ and $Q(x)\neq 0$ are polynomial functions.

Intercepts

Use the following rules to determine the intercepts, or zeros, of a rational function $\displaystyle f(x)=\frac{P(x)}{Q(x)}$.
  1. To determine the $x$-intercepts, set $f(x)=0$ and solve for $x$. This is equivalent to solving $P(x)=0$.
  2. To determine the $y$-intercepts, set $x=0$ and solve for $y$. If $Q(0)=0$, then the function $f(x)$ does not have a $y$-intercept.
Intercepts of rational functions.

Domain

Let $\displaystyle f(x) = \frac{P(x)}{Q(x)}$ be a rational function where $P(x)$ and $Q(x) \neq 0$ share no common factors. To determine the domain of $f(x)$:
  1. Fully factor $Q(x)$.
  2. If $(x-c)$ is a linear factor of $Q(x)$, then $x=c$ cannot be a part of the domain of $f(x)$. The line $x=c$ is a vertical asymptote of $f(x)$.
  3. Irreducible quadratic factors of $Q(x)$ will not give restrictions on the domain of $f(x)$, so they can be ignored.
Domain of rational functions.

Problem Set 3.5

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