1. Algebraic Fractions

Simplifying Algebraic Fractions

Algebraic fractions are expressions where both the numerator and denominator are polynomials. Simplifying these fractions involves factorizing and cancelling common factors.

• Factorization: Factorize both numerator and denominator fully before cancelling.

• Cancelling: Only cancel factors that appear in both numerator and denominator.

• Example: Simplify Factorize: , Cancel :

Multiplying and Dividing Fractions

• To multiply: Multiply numerators together and denominators together, then simplify.

• To divide: Invert the second fraction and multiply.

• Example: Invert and multiply: Factorize and cancel common factors.

Adding and Subtracting Fractions

• Find the Lowest Common Denominator (LCD).

• Rewrite each fraction with the LCD, then add or subtract numerators.

• Example: Factorize denominators, find LCD, rewrite, and combine.

2. Functions

Notation

• Function notation: denotes a function named with variable .

• Alternative notations: , .

Domain, Range, and Graph

The domain is the set of all possible input values () for which the function is defined. The range is the set of all possible output values ().

• To find the domain, consider restrictions such as division by zero or square roots of negative numbers.

• To find the range, sketch the graph or solve for in terms of .

• Example: For , domain , range (excluding if is not included).

Defining Functions

• Some mappings are not functions unless the domain is restricted.

• Example: is not a function for unless domain is .

Composite Functions

A composite function is formed by applying one function to the result of another, denoted .

• Order matters: in general.

• Example: If , , then .

Inverse Functions and Their Graphs

The inverse function reverses the effect of . The graph of is the reflection of in the line .

• To find the inverse: Replace with , swap and $y$, solve for $y$.

• Example: Swap: Solve: So

Modulus Functions

The modulus function returns the non-negative value of . Graphically, it reflects negative parts of a function above the $x$-axis.

• Example: To sketch , first sketch , then reflect any part below the -axis above it.

Standard Graphs

• Common graphs include , , , , and their transformations.

• Transformations include translations, reflections, stretches, and compressions.

3. Trigonometry (Overview from Table of Contents)

Trigonometric functions and identities are essential in calculus for solving equations and modeling periodic phenomena.

• Key functions: , ,

• Identities: Pythagorean, double angle, sum and difference, etc.

• Graphs: Know the shapes and key points of , , .

4. Differentiation (Preview from Table of Contents)

Differentiation is the process of finding the derivative, which represents the rate of change of a function.

• Chain Rule: Used for composite functions.

• Product Rule:

• Quotient Rule:

5. Exponentials and Logarithms (Preview from Table of Contents)

Exponentials and logarithms are used to model growth and decay, and to solve equations involving powers.

• Exponential function:

• Logarithmic function:

• Properties: ,

6. Numerical Methods (Preview from Table of Contents)

Numerical methods are used to approximate solutions to equations that cannot be solved algebraically.

• Root finding: Methods such as iteration and bisection.

• Convergence: Conditions under which iterative methods approach the correct solution.

Additional info:

• Some content previewed here (e.g., trigonometry, differentiation, exponentials, numerical methods) is inferred from the table of contents and not fully detailed in the provided images. For a complete study, refer to the full notes or textbook.