1. Rational Functions A rational function is a function of the form where g (x) 0

2. Domain of a Rational Function The domain of a rational function is the set of real numbers x so that g (x) 0 Example If the domain is all real numbers except x = 0

3. Domain of a Rational Function If then the domain is the set of all real numbers except x = -2 If then the domain is the set of all real numbers x > 5

4. Domain of a Rational Function then the domain is the set of all real numbers, because the denominator of this function is never equal to zero If

5. Unbounbed Functions A function f(x) is said to be unbounded in the positive direction if as x gets closer to zero, the values of f (x) gets larger and larger. We write this as f (x) as x 0 (read f (x) approaches infinity as x goes to 0)

6. An Unbounbed Function x -1/10500 -1/1005,0000 -1/10005,000,00 0 1/10005000,000 1/1005,0000 1/10500 Note that as x approaches 0, f (x) becomes very large The function below is unbounded

7. Unbounbed Functions Note that as x gets closer to zero, the values of f (x) gets smaller and smaller. We say f (x) is unbounded in the negative direction. We write this as f (x) as x 0 (read f (x) approaches negative infinity as x goes to 0) x -1/10100 -1/10010,000 -1/10001,000,000 1/10001,000,000 1/10010,000 1/10100

8. Unbounbed Functions If as x gets closer to zero from the left, the values of f (x) gets smaller and smaller, and as x gets closer to zero from the right, the values of f (x) gets larger and larger, we say f (x) is unbounded in both direction

9. Unbounbed Functions and write this as f (x) as x 0 ( read f (x) approaches positive or negative infinity as x goes to 0)

10. Unbounbed Functions

11. Vertical Asymptote For any rational function in lowest terms and a real number c so that h (c) 0 and g (c)= 0 the line x = c is called a vertical asymptote The function has a vertical asymptote X = 4

12. Vertical Asymptote For the function the vertical asymptote is x = -2 The function has no vertical asymptote because the denominator is never zero `

13. Horizontal Asymptote If the degree of the denominator of a rational function f (x) = h (x) / g (x) is greater than or equal to the degree of the numerator of the rational function, then f (x) has a horizontal asymptote.

14. Horizontal Asymptote If the degree of the denominator is greater than the of the degree of the numerator, then the horizontal asymptote is y = 0 Example y = 0 is the horizontal asymptote

15. Horizontal Asymptote If the degree of the denominator is equal to the of the degree of the numerator, then the horizontal asymptote is y = a where a is a non zero real number Example y = 2 is the horizontal asymptote

16. Horizontal Asymptote Horizontal asymptote is y = 2 Horizontal asymptote is y = 0 Examples of horizontal asymptotes

17. Slant Asymptote If the degree of the numerator is greater to the degree of the denominator, we have a slant asymptote Example of a function that has a slant asymptote

18. Please review problems solved in class before doing your homework