Rational Expressions

1. Rational Expressions
   A rational expression is an expression of the form , where and are polynomials, and . In particular, if is a nonzero constant, becomes a polynomial, so polynomials are also rational expressions.
   
   
   
2. A rational expression that is not a polynomial is called a fractional expression.
   
   
3. Properties of Rational Expressions
   For three polynomials , , and ( and ):


4. Arithmetic Operations of Rational Expressions
   For four polynomials , , , and ( and ):
1. Addition :
2. Subtraction :
3. Multiplication :
4. Division : with
5. Commutative and associative properties hold for addition and multiplication of rational expressions.
   
   
6. Calculations for Special Forms of Rational Expressions
1. When (the degree of the numerator) (the degree of the denominator), divide the numerator by the denominator to rewrite it as a sum of a polynomial and a rational expression where the degree of the numerator is less than that of the denominator.
2. For calculations involving more than four rational expressions, group pairs simplify the process.
3. If the denominator is a product of multiple factors, use partial fraction decomposition:
   (with ).
4. For rational expressions involving another rational expression in the numerator or denominator, simplify as follows:
   
5. Rational expressions containing another fractional expression in either the numerator or denominator are called nested fractions.

Rational Functions

1. Rational Functions
1. A function where is a rational expression in is called a rational function. If is a polynomial in , the function is called a polynomial function.
   
   
   
2. Domain of a Rational Function : If the domain is not specified, it is assumed to be all real numbers except where the denominator equals zero.
2. Graph of the Rational Function
1. The domain and range are all real numbers except zero.
2. If , the graph lies in the first and third quadrants.
   If , the graph lies in the second and fourth quadrants.
3. The graph is symmetric with respect to the origin and the lines and .
4. The asymptotes are the -axis ( ) and the -axis ( ).
5. As increases, the graph moves farther from the origin.




3. Graph of the Rational Function
   The graph of is the graph of translated by units in the -direction and units in the -direction.
1. Domain : real numbers such that
   Range : real numbers such that
2. The asymptotes are the lines and .
3. The graph is symmetric with respect to the point .
4. The graph is symmetric with respect to two lines and through with slopes .




4. Graph of the Rational Function
   The graph of is obtained by rewriting it in the form
1. Asymptotes : and
2. The graph is symmetric with respect to the point


5. Finding the Inverse of the Rational Function
   
   [Method 1]
1. Solve for , that is, rewrite it as .
2. Swap and to obtain .
[Method 2] Using the formula
6. The inverse of is , where the signs and positions of and are reversed.