P.5 Rational Expressions
I Domain
Domain:
Rational expressions :
Finding domain
a. polynomials:
b. Radicals: keep it real!
i. sqrt(x-2) x>=2 [2, inf)
ii. cubert(x-2) all reals since cube rootscan be positive and negative
c. rational expressions: can’t divide by zero (undefined) so look at denominator and set it equal to zero.
x +1 x2 +4 x + 3 i. ii. ()()x−2 x − 4 x + 3
II Simplifying rational expressions
*simplifying rational expression can have an impact on domain – helps determine graphs in this course and calculus.
x2 +8 x − 20 #36 x2 +11 x + 10
III Operatoins with Rational Expressions – remember fraction operations from P.1 notes
A) Multiply #54
B) Dividing #96
C) Combining rational expressions Ex 7
IV Complex Fractions
Complex fraction: separate fractions in numerator and/or denominator
x + 3 EX 4 2 3 2 − x
2 methods: straight division, multiply numerator and denominator by LCD
Factoring out negative exponents Ex 10
2 ways: factor out smallest exponent, multiply top/bottom by the smallest exponent with the opposite sign
V Difference Quotients: goal is to eliminate the original denominator
x+ h − x Ex 11 rationalize numerator h 4.1 Rational Functions and Asymptotes I Intro Rational function: given that and are polynomials. = Ex1 Domain of a rational function =
II Vertical and Horizontal Asymptotes Definitions p.333 blue box 1) The line = is a vertical Asymptote of the graph of if → ∞ or → −∞.
2) The line = is a horizontal asymptote of the graph of if → as → ∞ or → −∞.
Examples for domain (from p. 333) f(x)= =