Rational Functions and Cost Analysis

Cost Functions

Cost functions are used in business and economics to model the total cost of producing a certain number of items. They often include fixed and variable costs.

• Cost Function, C(x): The total cost of producing x items. Typically written as , where F is the fixed cost and V is the variable cost per item.

• Average Cost Function, \( \overline{C}(x) \): The average cost per item when producing x items.

• Application: To find the cost of producing a specific number of items, substitute the value into the cost function.

• Example: If , then the average cost for 10 items is .

Comparing Rental Costs

When comparing two rental options with different pricing structures, set up equations for each and solve for the break-even point.

• Example: Acme: ; Interstate:

• Set and solve for m (miles) to find when Acme is cheaper.

Rational Expressions

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplification involves factoring and reducing common factors.

• Key Steps:

  1. Factor numerator and denominator completely.

  2. Cancel any common factors.

  3. State restrictions (values that make the denominator zero).

• Example: simplifies to

Finding Common Denominators

To add or subtract rational expressions, find a common denominator, usually the least common multiple (LCM) of the denominators.

• Example: For and , factor denominators and find LCM.

Restrictions on Variables

Restrictions are values that make the denominator zero and must be excluded from the domain.

• Example: is undefined for .

Radical and Rational Exponents

Radical Notation and Rational Exponents

Expressions with rational exponents can be rewritten in radical form and vice versa.

• Key Formulas:

• Example:

Simplifying with Rational Exponents

• Example: can be written as

Solving Equations and Inequalities

Solving Polynomial and Radical Equations

To solve equations involving polynomials or radicals, isolate the variable and use appropriate algebraic techniques.

• Example: becomes , so

• Example: leads to , so

Absolute Value Equations

Absolute value equations have two possible solutions, one for the positive and one for the negative case.

• Example: gives or

Solving for a Variable (Literal Equations)

Literal equations involve solving for one variable in terms of others.

• Example: For , solve for :

• Example: For , solve for :

Inequalities and Interval Notation

Solving and Graphing Linear Inequalities

Linear inequalities are solved similarly to equations, but the solution is a range of values, often represented on a number line and in interval notation.

• Key Steps:

  1. Solve the inequality for the variable.

  2. Graph the solution on a number line.

  3. Write the solution in interval notation.

• Example: leads to ; interval notation:

Compound Inequalities

Compound inequalities involve two inequalities joined by 'and' or 'or'.

• Example: and

• Solve each separately, then find the intersection (for 'and') or union (for 'or').

Absolute Value Inequalities

Absolute value inequalities are solved by considering both the positive and negative cases.

• Example: leads to , so

Summary Table: Key Concepts