CALCULUS STUDY GUIDE

SECTION 1.1 - 1.2: Concept of Functions

This section introduces the foundational concept of functions, which is essential for understanding calculus. It covers the definition, domain, range, and various representations of functions.

• Definition of a Function: A function is a relation that assigns exactly one output to each input from a specified set. Formally, if is a function, then for every in the domain, there is a unique .

• Domain and Range: The domain is the set of all possible input values (), and the range is the set of all possible output values ().

• Function Notation: Functions are commonly written as , where is the input variable.

• Evaluating Functions: To evaluate a function, substitute the input value into the function's formula. Example: If , then .

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

SECTION 1.3: Basic Classes of Functions

This section classifies functions into several basic types, each with unique properties and applications in calculus.

• Linear Functions: Functions of the form , where is the slope and is the y-intercept.

• Quadratic Functions: Functions of the form , where , , and are constants.

• Polynomial Functions: Functions that are sums of powers of with constant coefficients, e.g., .

SECTION 1.4: Trigonometric Functions

Trigonometric functions are periodic functions that relate angles to ratios of sides in right triangles. They are fundamental in calculus for modeling periodic phenomena.

• Sine and Cosine: and are defined for all real and have ranges .

• Tangent: is undefined where .

• Periodicity: and have period ; has period .

• Example: , .

SECTION 1.5: Inverse Functions

Inverse functions reverse the effect of the original function, swapping inputs and outputs. They are crucial for solving equations and understanding function behavior.

• Definition: If is a function, its inverse satisfies and for all in the domain of $f^{-1}$.

• Graphical Interpretation: The graph of is the reflection of the graph of across the line .

• Finding Inverses: To find the inverse, solve for in terms of , then interchange $x$ and $y$.

• Example: If , then .

SECTION 1.6: Graphs of Inverse Functions

This section explores how the graphs of inverse functions relate to the original functions, emphasizing symmetry and graphical techniques.

• Symmetry: The graph of an inverse function is symmetric to the original function's graph with respect to the line .

• Verifying Inverses: Check that and for sample values.

• Example: The graph of and (for ) are inverses and symmetric about .