Rational Functions A rational function is a function of the form where g (x) 0
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
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
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
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)
An Unbounbed Function x -1/ /1005, /10005,000,00 0 1/ ,000 1/1005,0000 1/10500 Note that as x approaches 0, f (x) becomes very large The function below is unbounded
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/ /10010,000 -1/10001,000,000 1/10001,000,000 1/10010,000 1/10100
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
Unbounbed Functions and write this as f (x) as x 0 ( read f (x) approaches positive or negative infinity as x goes to 0)
Unbounbed Functions
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
Vertical Asymptote For the function the vertical asymptote is x = -2 The function has no vertical asymptote because the denominator is never zero `
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.
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
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
Horizontal Asymptote Horizontal asymptote is y = 2 Horizontal asymptote is y = 0 Examples of horizontal asymptotes
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
Please review problems solved in class before doing your homework