College Algebra Review: Simplification, Functions, Radicals, and Applications

1. Simplifying Algebraic Expressions

Simplifying Rational and Radical Expressions

Algebraic simplification involves reducing expressions to their simplest form by combining like terms, factoring, and canceling common factors.

• Rational Expressions: Expressions involving fractions with polynomials in the numerator and denominator. Simplify by factoring and reducing common factors.

• Radical Expressions: Expressions containing roots (such as square roots or cube roots). Simplify by extracting perfect powers and rationalizing denominators if necessary.

• Example: Simplify $\frac{x^2-1}{x^2+1}$ by factoring numerator as $(x-1)(x+1)$ and checking for common factors with the denominator.

2. The Difference Quotient

Definition and Application

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is foundational for calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Steps:

  1. Substitute $x+h$ into the function to find $f(x+h)$.

  2. Subtract $f(x)$ from $f(x+h)$.

  3. Divide the result by $h$.

• Example: For $f(x) = 2x^2 + 5x$, compute $f(x+h)$, then apply the difference quotient formula.

3. Solving Rational Equations

Equations Involving Fractions

To solve rational equations, find a common denominator, multiply both sides to clear denominators, and solve the resulting equation.

• Key Steps:

  1. Identify the least common denominator (LCD).

  2. Multiply both sides by the LCD to eliminate denominators.

  3. Solve the resulting linear or quadratic equation.

  4. Check for extraneous solutions by substituting back into the original equation.

• Example: Solve $\frac{2}{x-9} - \frac{3}{x+3} = 1$ by finding the LCD $(x-9)(x+3)$.

4. Solving Rational Inequalities

Finding Solution Sets

Rational inequalities involve expressions with variables in the denominator. The solution set is found by determining where the expression is greater than, less than, or equal to a given value.

• Key Steps:

  1. Move all terms to one side to set the inequality to zero.

  2. Find critical points by setting the numerator and denominator to zero.

  3. Test intervals between critical points to determine where the inequality holds.

  4. Express the solution in interval notation.

• Example: Solve $\frac{x-5}{x+1} > 5$.

5. Function Evaluation

Substituting Values into Functions

To evaluate a function at a given value, substitute the value for the variable in the function's formula.

• Example: For $f(x) = \sqrt{5-3x}$, find $f(1)$ and $f(2)$ by substituting $x=1$ and $x=2$.

• For $g(x) = \frac{1}{\sqrt{x}}$: Evaluate $g(4)$ and $g(1)$.

6. Domain of Functions

Finding the Domain and Expressing in Interval Notation

The domain of a function is the set of all input values (x-values) for which the function is defined.

• For square root functions: The radicand must be non-negative.

• For rational functions: The denominator must not be zero.

• Interval Notation: Use parentheses for excluded values and brackets for included values.

• Example: For $g(x) = \frac{1}{\sqrt{x}}$, the domain is $x > 0$, or $(0, \infty)$.

7. Simplifying Radicals and Exponents

Working with Roots and Exponential Expressions

Radical expressions can often be simplified by factoring out perfect powers and using exponent rules.

• Key Properties:

  • $\sqrt[n]{a^m} = a^{m/n}$

  • Negative exponents: $a^{-n} = \frac{1}{a^n}$

  • Product and quotient rules for exponents

• Example: Simplify $\sqrt[3]{-8y^6}$ as $-2y^2$.

8. Expressing in Simplest Radical Form

Combining and Simplifying Radicals

Expressing answers in simplest radical form involves extracting perfect squares or cubes and combining like terms.

• Example: $\sqrt{12x^3y^2z^5}$ can be simplified by factoring out perfect squares from each variable.

9. Operations with Radicals

Adding, Subtracting, and Multiplying Radicals

Radicals can be combined if they have the same index and radicand. Multiplication and division follow the distributive property and exponent rules.

• Example: $3\sqrt{4} + 4\sqrt{8x^2}$ can be simplified by evaluating the square roots and combining like terms.

10. Word Problems: Rates and Motion

Setting Up and Solving Real-World Problems

Word problems often involve translating a scenario into an algebraic equation, then solving for the unknown.

• Rate Problems: Use the formula $\text{Distance} = \text{Rate} \times \text{Time}$.

• Example: If Vanessa hikes 2 mph slower than Xavier and their times and distances are given, set up equations to solve for their speeds.

• Additional info: For problems involving total travel time and different speeds, set up equations for each segment and solve for the unknown variable.