1. Section 5.2 – Properties of Rational Functions
Defn:
Rational Function
A function in the form: 𝑅 𝑥 = 𝑝(𝑥) 𝑞(𝑥)
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 𝑥 2 −4 𝑥+5
ℎ 𝑥 = 2 𝑥 2 −4
𝑔 𝑥 = 𝑥 2 −1 𝑥−1

2. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
𝑓 𝑥 = 𝑥 2 −3 𝑥+4
𝑥+4=0
𝑥=−4
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –4} or
(-, -4)  (-4, )

3. graph

4. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
𝑔 𝑥 = 𝑥 2 −4 𝑥−2
𝑥−2=0
𝑥=2
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  2} or
(-, 2)  (2, )

5. graph

6. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
ℎ 𝑥 = 2 𝑥 2 −9
𝑥 2 −9=0
(𝑥−3)(𝑥+3)=0
x=−3, 3
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –3, 3} or
(-, -3)  (-3, 3)  (3, )

7. graph

8. Section 5.2 – Properties of Rational Functions
Domain of a Rational Function
ℎ 𝑥 = 2𝑥+3 𝑥 2 −2𝑥−15
𝑥 2 −2𝑥−15=0
(𝑥−5)(𝑥+3)=0
x=−3, 5
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –3, 5} or
(-, -3)  (-3, 5)  (5, )

9. Section 5.2 – Properties of Rational Functions
Linear Asymptotes (vertical, horizontal, or oblique)
Lines in which a graph of a function will approach. By approach we mean each successive value of X puts the graph closer to the asymptote than the previous value.
Vertical Asymptote
A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero, in this case the factors would cancel.
Example
𝑓 𝑥 = 𝑥 2 −16 𝑥+5
= (𝑥−4)(𝑥+4) 𝑥+5
x=−5
A vertical asymptotes exists at x = -5.
VA: 𝑥=−5

10. graph

11. Section 5.2 – Properties of Rational Functions
Asymptotes
Vertical Asymptote
Example
𝑓 𝑥 = 𝑥 2 −𝑥−6 𝑥 2 −7𝑥+12
= (𝑥+2)(𝑥−3) (𝑥−4)(𝑥−3)
x=3, 4
A vertical asymptote exists at x = VA: 𝑥=4
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.

12. graph

13. Section 5.2 – Properties of Rational Functions
Asymptotes
Horizontal Asymptote
A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, or
if the largest exponent in the denominator is larger than the largest exponent in the numerator.
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.
If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is 𝑦=0.

14. Section 5.2 – Properties of Rational Functions
Asymptotes
Horizontal Asymptote
Example
𝑓 𝑥 = 5𝑥 3 −2 𝑥 2 −7 2𝑥 3 −7𝑥+10
HA: 𝑦= 5 2
A horizontal asymptote exists at y = 5/2.
𝑓 𝑥 = 𝑥−6 𝑥 2 −7𝑥+12
A horizontal asymptote exists at y = 0.
HA: 𝑦=0

15. graph

16. Section 5.2 – Properties of Rational Functions
Asymptotes
Oblique (slant) Asymptote
An oblique asymptote exists if the largest exponent in the numerator is one degree larger than the largest exponent in the denominator.
**Note**
Other non-linear asymptotes can exist for a rational function.

17. Section 5.2 – Properties of Rational Functions
Asymptotes
Oblique Asymptote
Example
𝑓 𝑥 = 𝑥 2 +1 𝑥
An oblique asymptote exists.
Long division is required.
We ignore the remainder if it exists 𝑥 −
𝑥 2 0𝑥
An oblique asymptote exists at y = x. OA: 𝑦=𝑥

18. graph

19. Section 5.2 – Properties of Rational Functions
Asymptotes
Oblique Asymptote
Example
𝑓 𝑥 = 4𝑥 4 + 2𝑥 2 +𝑥−1 2𝑥 3 +3𝑥
An oblique asymptote exists.
Long division is required. 2𝑥 −
4𝑥 4 −
6𝑥 2
An oblique asymptote exists at y = 2x
−4𝑥 2
OA: 𝑦=2𝑥

20. graph