8. Rational Functions
Example: Finding the Domain of Rational Functions
Find the domain of the following rational functions by
determining the values for which each denominator is zero, thus
making the rational expression undefined.
a. π π₯ =
2
π₯2
b. π π₯ =
π₯+1
2π₯+1
c. π¦ =
3π₯+4
π₯2+π₯β12
14. 3. π =
πβπ
ππβπ
A rational function
can have numerous
vertical asymptotes,
at most one
horizontal
asymptote or
oblique (slant)
asymptote.
15. 4. π =
ππ
ππβπ
A rational function
can have numerous
vertical asymptotes,
at most one
horizontal
asymptote or
oblique (slant)
asymptote.
16. To determine the asymptotes of a rational function
π π₯ =
π(π₯)
π(π₯)
=
πππ₯π
+ ππβ1π₯πβ1
+ β¦ + π1 π₯ + π0
πππ₯π + ππβ1π₯πβ1 + β¦ + π1 π₯ + π0
Follow these guidelines:
1. A graph has a vertical asymptote, line x=a, at each value a where the
denominator is zero.
2. The horizontal asymptote is determined using the following:
a. If n < m, then the graph has a horizontal asymptote at line y = 0.
b. If n = m, then the graph has a horizontal asymptote at line y =
ππ
ππ
.
c. If n > m, then the graph has no horizontal asymptote.
3. The graph of a rational function has an oblique asymptote if n = m + 1.
To determine the equation of the oblique asymptote, the numerator P(x)
is divided by the denominator Q(x) using long division. In the resulting
quotient
π(π₯)
π(π₯)
+ π· π₯ =
π (π₯)
π(π₯)
, π¦ = π·(π₯) is the equation of the oblique
asymptote.
17. Determine the asymptotes (vertical, horizontal, oblique), if any, of the
given rational functions.
a. π π₯ =
π₯β3
π₯2β9
b. π¦ =
5π₯β2π₯2β3
π₯2+4π₯+4
c. π¦ =
π₯3+2π₯2βπ₯β2
π₯2+1
18. Graphing a Rational Function
Follow these steps in graphing a rational function f(x):
1. Simplify the faction by factoring the numerator and
denominator, if possible.
2. Find and plot the x-intercept(s) by solving f(x) = 0.
3. Find and plot the y-intercept by evaluating f(0).
4. Determine and sketch the graph of the asymptotes.
5. Plot points between and beyond x-intercept and vertical
asymptotes to approximate the behavior of the graph in each
interval.
6. Draw smooth curves to connect the points and to sketch how
the graph behaves as the values get closer to the asymptotes.
19. Example: Graphing Rational Functions
Graph the following rational functions, then determine their domain
and range.
a. π¦ =
4βπ₯
2π₯+1
b. π¦ =
π₯3β2π₯2β15π₯
π₯2β2π₯
20. Rational Equations
β’ A rational equation is an equation involving one or more
rational expressions.
β’ Examples:
1.
2
π₯β3
= 0
2.
1
π₯
β
π₯
π₯+1
= 4
3.
π₯+1
π₯2β3π₯+2
=
5
2π₯
21. Solving Rational Equations
Procedure in solving rational equations:
a. Find the LCD
b. Multiply both sides by the LCD
c. Check the solutions by substitution
26. Rational Inequalities
β’ A rational inequality is an inequality involving at least one
rational expression.
β’ Examples:
1.
π₯
π₯+1
β₯ 1
2.
1
π₯
β
2
π₯2 < 0
3.
π₯2+2π₯+1
π₯β4
β€
π₯
1βπ₯
27. Solving Rational Inequalities
Here are the steps in solving rational inequality:
1. Express the inequality in general form, that is rewrite it such
that one side of the inequality is zero.
2. Find the LCD of the