Rational Functions: Operations and Equations

Rational Functions

Rational functions are mathematical expressions consisting of a ratio of two polynomial functions. They can be written as f(x) = p(x)/q(x), where p(x) is the numerator, q(x) is the denominator, and both polynomials have degree one or higher. These functions play a significant role in calculus and algebraic manipulation.

Operations with Rational Functions

Operations with rational functions involve performing arithmetic operations such as addition, subtraction, multiplication, division, and composition. These operations are performed similarly to those of polynomial functions and follow certain rules. For example:

Addition and Subtraction

To add or subtract two rational functions, their numerators are combined, and their denominators are combined, and the result is the sum or difference of the original functions.

Multiplication

To multiply two rational functions, the numerators are multiplied, and the denominators are multiplied, and the result is the quotient of the original functions.

Division

To divide two rational functions, the denominator of the second function is multiplied by the numerator of the first function, and the result is the quotient of the original functions.

Solving Rational Equations

Solving rational equations involves isolating the variable on one side of the equation and then simplifying the resulting expression by performing arithmetic operations. This process includes the following steps:

1. Step 1: Multiply both sides of the equation by the least common multiple of the denominators to clear the fractions.
2. Step 2: Combine the numerators on one side of the equation.
3. Step 3: Perform operations to isolate the variable on one side of the equation.
4. Step 4: Simplify the resulting expression by performing arithmetic operations.

For example, consider the equation (3x^2 - 2x + 1) / (2x - 1) = 0. Here is how the solution would proceed:

1. Multiply both sides of the equation by (2x - 1) to clear the fraction: (2x - 1) * (3x^2 - 2x + 1) / (2x - 1) = 0 multiplies to 3x^2 - 2x + 1 = 0.
2. Combine the numerators on one side of the equation: (3x^2 - 2x + 1) = 0.
3. Perform operations to isolate the variable on one side of the equation: 3x^2 - 2x + 1 = 0 can be factored to (3x - 1)(x - 1) = 0.
4. Simplify the resulting expression by performing arithmetic operations: (3x - 1)(x - 1) = 0 implies 3x - 1 = 0 or x - 1 = 0, which gives x = 1/3 or x = 1 as the possible solutions.