College Algebra: Rational Expressions, Functions, and Inequalities Study Guide

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 $C(x) = F + Vx$, 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. $\overline{C}(x) = \frac{C(x)}{x}$

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

• Example: If $C(x) = 1000 + 50x$, then the average cost for 10 items is $\overline{C}(10) = \frac{1000 + 50 \times 10}{10} = 150$.

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: $C_A = 4d + 0.15m$; Interstate: $C_I = 3d + 0.08m$

• Set $C_A < C_I$ 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: $\frac{x^2 - 7x - 15}{x^2 - 25}$ simplifies to $\frac{(x-10)(x+3)}{(x-5)(x+5)}$

Finding Common Denominators

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

• Example: For $\frac{x+5}{7x^2+21x}$ and $\frac{5}{x^2+6x+9}$, factor denominators and find LCM.

Restrictions on Variables

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

• Example: $\frac{x-9}{x+1}$ is undefined for $x = -1$.

Radical and Rational Exponents

Radical Notation and Rational Exponents

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

• Key Formulas:

  • $a^{m/n} = \sqrt[n]{a^m}$

  • $\sqrt[n]{a} = a^{1/n}$

• Example: $25^{3/2} = (\sqrt{25})^3 = 5^3 = 125$

Simplifying with Rational Exponents

• Example: $\sqrt[3]{15ab}$ can be written as $(15ab)^{1/3}$

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: $x^3 - 8 = -4$ becomes $x^3 = 4$, so $x = \sqrt[3]{4}$

• Example: $\sqrt{y+1} - 9 = 0$ leads to $y+1 = 81$, so $y = 80$

Absolute Value Equations

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

• Example: $|x+1| = 7$ gives $x+1 = 7$ or $x+1 = -7$

Solving for a Variable (Literal Equations)

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

• Example: For $A = \frac{1}{2}bh$, solve for $h$: $h = \frac{2A}{b}$

• Example: For $A = \frac{4}{\pi}r^2$, solve for $r$: $r = \sqrt{\frac{A}{4\pi}}$

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: $3x - 5 < -20$ leads to $x < -5$; interval notation: $(-\infty, -5)$

Compound Inequalities

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

• Example: $x - 2 < 6$ and $3x + 6 < 21$

• 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: $|x-1| \leq 3$ leads to $-3 \leq x-1 \leq 3$, so $-2 \leq x \leq 4$

Summary Table: Key Concepts

Concept	Key Formula/Fact	Example
Cost Function	$C(x) = F + Vx$	$C(10) = 1000 + 50 \times 10 = 1500$
Average Cost	$\overline{C}(x) = \frac{C(x)}{x}$	$\overline{C}(10) = 150$
Rational Expression	Factor and reduce	$\frac{x^2-9}{x+3} = x-3$ (for $x \neq -3$)
Radical Exponent	$a^{m/n} = \sqrt[n]{a^m}$	$8^{2/3} = (\sqrt[3]{8})^2 = 4$
Interval Notation	$(a, b)$ or $[a, b]$	$x > 2$ is $(2, \infty)$