Functions and Their Properties

Definition and Evaluation of Functions

A function is a rule that assigns to each input exactly one output. Functions can be represented algebraically, graphically, or numerically.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Evaluating Functions: Substitute the input value into the function's formula to find the output.

Example: If , then .

Average Velocity and Rates of Change

Average Velocity

The average velocity of an object over an interval is the change in position divided by the change in time.

• Formula: , where is the position function.

• Represents the slope of the secant line between two points on the position-time graph.

Example: If , the average velocity from to is .

Limits

Definition of a Limit

The limit of a function as approaches is the value that gets closer to as gets closer to .

• Notation:

• If approaches as approaches , then .

Example: can be evaluated by factoring and simplifying.

One-Sided Limits

One-sided limits consider the behavior of a function as approaches from the left () or right ().

• Left-hand limit:

• Right-hand limit:

Example: For a piecewise function, the left and right limits at a point may differ.

Evaluating Limits Algebraically

• Direct substitution: If is continuous at , substitute into .

• Factoring: Factor numerator and denominator to simplify and remove discontinuities.

• Rationalization: Multiply by a conjugate to simplify expressions with square roots.

Example: can be evaluated by multiplying numerator and denominator by .

Continuity

Definition of Continuity

A function is continuous at if:

• is defined.

• exists.

• .

If any of these conditions fail, the function is discontinuous at .

Example: For , check continuity at by evaluating the limit and the function value.

Types of Discontinuities

• Removable discontinuity: The limit exists, but is not defined or does not equal the limit.

• Jump discontinuity: The left and right limits exist but are not equal.

• Infinite discontinuity: The function approaches infinity near .

Asymptotes

Vertical and Horizontal Asymptotes

An asymptote is a line that a graph approaches but never touches.

• Vertical asymptote: Occurs at where approaches infinity as approaches .

• Horizontal asymptote: Occurs when approaches a constant value as approaches infinity.

Example: For , is a vertical asymptote.

Piecewise Functions and Graphical Analysis

Piecewise Functions

A piecewise function is defined by different expressions on different intervals of the domain.

• Evaluate limits and continuity at the points where the formula changes.

• Check left and right limits at transition points.

Example:

Graphical Interpretation

Graphs can be used to estimate limits, identify discontinuities, and locate asymptotes.

• Look for jumps, holes, or vertical asymptotes.

• Use the graph to find , , and .

Steady-State Solutions

Steady-State Value

A steady-state solution is a constant value that a system approaches as time goes to infinity.

• Often found by evaluating the limit as .

• Used in modeling population, chemical reactions, and other dynamic systems.

Example: If models population, the steady-state is .

Summary Table: Types of Discontinuities

Additional info: These notes expand on the original homework questions by providing definitions, formulas, and examples for key Calculus I concepts: functions, limits, continuity, asymptotes, and steady-state solutions. The table summarizes types of discontinuities for quick reference.