Rational Functions and inequalities

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Example 1*: Find the Domain of a Rational Function

Find the domain of the following rational functions

Example 2: Graph the following functions

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Section 4.2, 4.3 & 4.4
Section 4.2: Properties of Rational Functions
https://mediaplayer.pearsoncmg.com/assets/1NIzrJsdy0qx5jwM_vgh0NyjCAMp7Iuu
Author in Action: Find the Domain of a Rational Function (5:56)
Post 1
Definition: A rational function is a function of the form 𝑅(𝑥 ) =

p
𝑞

where p and q are polynomial

functions and q is not the zero function. The domain of a rational function is
_______________________ except those for which the denominator q is _0.___.
WARNING: The domain of a rational function must be found before writing the function in its lowest
terms.
Example 1*: Find the Domain of a Rational Function
Find the domain of the following rational functions
𝑥2 − 4
(
)
(𝑎)* 𝑅 𝑥 =
𝑥+4
(-∞, -4) U (-4, ∞)

(𝑐)* 𝑅 (𝑥 ) =

𝑥−5
𝑥2 + 2

(𝑏)* 𝑅(𝑥 ) =

𝑥+6
𝑥 2 + 8𝑥 + 12

(-∞, -6) U (-6, -2)U (-2, ∞)

(𝑑) 𝑅(𝑥 ) =

(-∞, ∞)

𝑥2 − 9
3

(𝑒) 𝑅(𝑥 ) =

(-∞, ∞)

(-∞, -2) U (-2, ∞)

End of Post 1

https://mediaplayer.pearsoncmg.com/assets/EKTJNnlu3ciT871DoeoWkYscimlBFGJD
Author in Action: Graph Transformations of a Rational Function (18:10)
Example 2: Graph the following functions
1
y=
x

𝑥2 − 4
𝑥+2

y=

1
x2

Rational Functions

In Section 2.4 we created our library of functions and in Section 2.5 we transformed those basic
1
functions. We can do the same types of transformations with the basic rational functions y = and
x
1
y= 2 .
x

Post 2
Example 3*: Graph Transformations of a Rational Function
1
+ 2 using transformations.
Graph the rational function R ( x ) =
2
( x − 3)

End of Post 2

Section 4.2, 4.3 & 4.4
Long-run behavior: As we learned in Section 51, the long-run behavior of a function concerns what
happens as the inputs get extreme ( 𝑥 → ±∞). As we saw before with polynomials, the leading terms
of 𝑝(𝑥) and 𝑞(𝑥) determine the long run behavior of r ( x ) .
Exploration 1: Long-Run Behavior and Vertical Asymptotes
3x + 1
1. Use a graphing utility to graph f ( x ) =
.
x +1
(a) What is happening to the function’s values as the x values move towards positive infinity?
How can you use mathematical notation to write this?
The function’s values approach to the number 3.
𝑎𝑠 𝑥 → ∞, 𝑓 (𝑥) → 3
(b) What is happening to the function’s values as the x values move towards negative infinity?
How can you use mathematical notation to write this?
The function’s values approach to 3.
𝑎𝑠 𝑥 → −∞, 𝑓 (𝑥 ) →3
(c) Look at the equation for 𝑓(𝑥). Can you explain why the function’s values get closer and
closer to the values found above as the x values approach positive and negative infinity?
Because y = 3 is the horizontal asymptote of the function.

Rational Functions
Know these definitions.

Horizontal and Vertical Asymptotes
Let R denote a function.
If, as x approaches infinity or as x approaches negative infinity, the values of R( x) approach some
fixed number L, then the line y = L is a horizontal asymptote of the graph of R. [Refer to (a) and (b)
below].
If, as x approaches some number c, the values increase or decrease indefinitely , then the line y = c is a
vertical asymptote of the graph of R. The graph of R never intersects a vertical asymptote. [Refer to
(c) and (d)].

Can the graph of a function intersect a horizontal asymptote?
Yes.
(d) What is the equation for the horizontal asymptote of f ( x ...

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