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Complete the Week 7 guided Lecture Notes document

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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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## Explanation & Answer

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Section 4.2, 4.3 & 4.4

Section 4.2: Properties of Rational Functions

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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

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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

Copyright © 2016 Pearson Education, Inc.

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

Copyright © 2016 Pearson Education, Inc.

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.

Copyright © 2016 Pearson Education, Inc.

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 ...