BackCalculus Midterm Study Guide: Functions, Limits, and Continuity
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Functions and Their Properties
Domain and Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce.
Example: For , the domain and range are both since any real number can be input and output.
Piecewise Functions
A piecewise function is defined by different expressions depending on the interval of the input variable.
Example:
Asymptotes
An asymptote is a line that a graph approaches but never touches. Rational functions often have vertical and horizontal asymptotes.
Example: For :
Vertical asymptote: (where denominator is zero)
Horizontal asymptote: (degree of numerator equals degree of denominator; leading coefficients ratio)
Inverse Functions
The inverse function reverses the effect of . If , then .
Graphing: The graph of an inverse function is a reflection of the original function across the line .
Trigonometric Functions and Identities
Trigonometric Identities
A trigonometric identity is an equation involving trigonometric functions that is true for all values in the domain.
Example: is an identity if it holds for all .
Trigonometric Function Values
Trigonometric functions (sin, cos, tan) have specific values for standard angles (0, , , , ).
$0$ | |||||
|---|---|---|---|---|---|
$0$ | $1$ | ||||
$1$ | $0$ | ||||
$0$ | $1$ | undefined |
Inverse Trigonometric Functions
Inverse trigonometric functions return the angle whose trigonometric function value is a given number.
Example: is equivalent to for .
Transformations of Functions
Horizontal Shifts
A horizontal shift of a function by units is represented by . This shifts the graph to the left by units.
Example: The graph of is the graph of shifted 3 units to the left.
Combining Functions
Function Composition
Functions can be combined by addition, subtraction, multiplication, division, or composition.
Example: If and , then .
Exponent and Logarithm Rules
Exponent Rules
Exponent rules allow simplification of expressions involving powers.
Example:
Exponential Functions
An exponential function models growth or decay. The general form is .
Example: If a population doubles every 24 hours, . After 3 days (), .
Logarithm Properties
Logarithms have properties that allow simplification:
Example: If and , then .
Limits and Continuity
Limits
The limit of a function describes the behavior as the input approaches a certain value.
Example: can be proven by choosing .
Evaluating Limits:
Continuity
A function is continuous on an interval if there are no breaks, jumps, or holes in its graph.
Example: A polynomial function is continuous everywhere; rational functions are continuous except where the denominator is zero.
One-to-One Functions and Inverses
Intervals of One-to-One Functions
A function is one-to-one if each output is produced by exactly one input. This property is necessary for a function to have an inverse.
Example: is one-to-one on intervals where the function is strictly increasing or decreasing, such as .
Summary Table: Key Concepts
Concept | Definition | Example |
|---|---|---|
Domain | Set of input values | for |
Range | Set of output values | for |
Asymptote | Line approached by graph | , for |
Inverse Function | Reverses original function | |
Limit | Value as input approaches point | |
Continuity | No breaks in graph | Polynomial functions |
Additional info: Some examples and explanations have been expanded for clarity and completeness.
