Section 5.2 – Properties of Rational FunctionsDefn: Rational Function

A function in the form:

The functions p and q are polynomials.The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

𝑓 (𝑥 )=2 𝑥2−4

𝑥+5h (𝑥 )= 2

𝑥2−4𝑔 (𝑥 )= 𝑥2−1

𝑥−1

Section 5.2 – Properties of Rational Functions Domain of a Rational Function

𝑓 (𝑥 )=𝑥2−3𝑥+4

𝑥+4=0𝑥=−4𝐷𝑜𝑚𝑎𝑖𝑛 :

or(-, -4) (-4, )

{x | x –4}

Section 5.2 – Properties of Rational Functions Domain of a Rational Function

𝑔 (𝑥 )= 𝑥2−4𝑥−2

𝑥−2=0

𝑥=2

𝐷𝑜𝑚𝑎𝑖𝑛 :

or

(-, 2) (2, )

{x | x 2}

Section 5.2 – Properties of Rational Functions Domain of a Rational Function

h (𝑥 )= 2

𝑥2−9

𝑥2−9=0(𝑥−3)(𝑥+3)=0x=−3 ,3𝐷𝑜𝑚𝑎𝑖𝑛 :

or

(-, -3) (-3, 3) (3, )

{x | x –3, 3}

Section 5.2 – Properties of Rational Functions Domain of a Rational Function

h (𝑥 )= 2𝑥+3𝑥2−2 𝑥−15

𝑥2−2 𝑥−15=0(𝑥−5)(𝑥+3)=0x=−3 ,5𝐷𝑜𝑚𝑎𝑖𝑛 :

or

(-, -3) (-3, 5) (5, )

{x | x –3, 5}

Linear AsymptotesLines in which a graph of a function will approach.

Vertical AsymptoteA vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero.

Example

Section 5.2 – Properties of Rational Functions

𝑓 (𝑥 )=𝑥2−16𝑥+5

¿(𝑥−4 )(𝑥+4)

𝑥+5 x=−5

A vertical asymptotes exists at x = -5. VA:

AsymptotesVertical Asymptote

Example

Section 5.2 – Properties of Rational Functions

𝑓 (𝑥 )= 𝑥2−𝑥−6𝑥2−7𝑥+12

¿(𝑥+2)(𝑥−3)(𝑥−4 )(𝑥−3)

x=3 ,4

A vertical asymptote exists at x = 4. VA:

A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero.

A hole exists in the graph at x = 3.

Horizontal AsymptoteA horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal,

Section 5.2 – Properties of Rational FunctionsAsymptotes

If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is .

If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients.

orif the largest exponent in the denominator is larger than the largest exponent in the numerator.

AsymptotesHorizontal Asymptote

Example

Section 5.2 – Properties of Rational Functions

𝑓 (𝑥 )= 𝑥−6𝑥2−7𝑥+12

A horizontal asymptote exists at y = 0.

𝑓 (𝑥 )=5 𝑥3−2𝑥2−7

2 𝑥3−7𝑥+10

A horizontal asymptote exists at y = 5/2. HA:

HA:

Oblique (slant) AsymptoteAn oblique asymptote exists if the largest exponent in the numerator is one degree larger than the largest exponent in the denominator.

Section 5.2 – Properties of Rational FunctionsAsymptotes

Other non-linear asymptotes can exist for a rational function.

**Note**

AsymptotesOblique Asymptote

Example

Section 5.2 – Properties of Rational Functions

𝑓 (𝑥 )=𝑥2+1𝑥

An oblique asymptote exists.

An oblique asymptote exists at y = x. OA:

Long division is required.

102 xxx𝑥

𝑥2

0−

0 𝑥

AsymptotesOblique Asymptote

Example

Section 5.2 – Properties of Rational Functions

𝑓 (𝑥 )=4 𝑥4+2 𝑥2+𝑥−12 𝑥3+3𝑥

An oblique asymptote exists.

An oblique asymptote exists at y = 2x

Long division is required.

120432 2343 xxxxxx2 𝑥

4 𝑥4− 6 𝑥2−−4 𝑥2

OA: