Functions & Continuity Definitions Flashcards

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Questions and Answers

What does 'domain' refer to in functions?

Input (x-values)

What does 'range' refer to in functions?

Output (y-values)

What is a 'one-to-one' function?

Each input has exactly one unique output.

What does 'onto' mean regarding functions?

Each output has an input. Signup and view all the answers

What characterizes an 'onto/one-to-one' function?

Each input has exactly one output and each output has exactly one unique input. Signup and view all the answers

What does 'discrete' mean in mathematics?

Finite set of points. The points are not connected. Signup and view all the answers

What does 'continuous' mean in terms of functions?

Infinite set of points. The points are connected and go on forever. Signup and view all the answers

What is the 'Vertical Line Test'?

If any vertical line passes through the graph more than once, the graph is NOT a function. Signup and view all the answers

What is 'function notation'?

Replaces y. Looks like f(x). Signup and view all the answers

What does it mean to 'evaluate' a function?

Replaces x (input) with a value or an expression. The 'Plug and Chug'. Signup and view all the answers

What is the formula for a 'linear function'?

f(x) = mx + b Signup and view all the answers

What is a 'line of symmetry'?

Vertical line, 'folding point' Signup and view all the answers

What is a 'turning point' on a graph?

When the graph changes direction. Signup and view all the answers

What is a 'relative minimum' on a graph?

Occurs when the graph changes from decreasing to increasing. Signup and view all the answers

What is a 'relative maximum' on a graph?

When a graph changes from increasing to decreasing. Signup and view all the answers

What are 'zeros' in relation to a graph?

Where the graph crosses the x-axis. Signup and view all the answers

What is an 'x-intercept'?

Graph crosses the x-axis. Signup and view all the answers

What is a 'y-intercept'?

Graph crosses the y-axis. Signup and view all the answers

What does 'positive' signify in a graph?

Where the graph lies above the x-axis. Signup and view all the answers

What does 'negative' signify in a graph?

Where the graph lies below the x-axis. Signup and view all the answers

What does 'increasing' mean regarding a function?

As the x-values increase, the value of the function increases. Signup and view all the answers

What does 'decreasing' mean regarding a function?

As the x-values increase, the value of the function decreases. Signup and view all the answers

What is 'end behavior'?

Describes how the values of the function behave as x goes to infinity and as f(x) goes to infinity. Signup and view all the answers

What does 'continuity' refer to in functions?

Connected lines or connected smooth curves. Signup and view all the answers

What are 'extrema' in calculus?

Maximum or minimum. Signup and view all the answers

What does 'symmetry' mean concerning graphs?

A vertical line that divides the graph of the function into 2 halves that are mirror images of one another. Signup and view all the answers

Study Notes

Functions and Continuity Concepts

Domain: Refers to the set of all possible input values (x-values) for a function.
Range: Represents all possible output values (y-values) produced by a function.

Characteristics of Functions

One-to-one: Each input has a unique output; no two different inputs share the same output.
Onto: Every possible output is associated with at least one input from the domain.
Onto/one-to-one: Each input has a unique output, and each output corresponds to exactly one input, ensuring a bijective relationship.

Types of Functions

Discrete: Consists of a finite number of separate points that are not connected.
Continuous: Contains an infinite number of points that are connected, extending indefinitely in both directions.

Graphical Analysis

Vertical Line Test: A method to determine if a curve is a function; if any vertical line intersects the graph more than once, it is not a function.
Function Notation: Defined by replacing y with f(x); a way to express functions clearly.
Evaluate: The process of substituting a specific value or expression for x in a function, often referred to as "Plug and Chug."

Linear Function

Linear Function Equation: Represented by f(x) = mx + b, where m is the slope and b is the y-intercept.

Graph Characteristics

Line of Symmetry: A vertical line that acts as a folding point, dividing the graph into two mirror-image halves.
Turning Point: The point at which a graph changes direction, indicating local extrema.
Relative Minimum: A point where the graph transitions from decreasing to increasing, representing a local low.
Relative Maximum: Occurs when the graph changes from increasing to decreasing, indicating a local high.

Intercepts and Axes

Zeros: Points where the graph intersects the x-axis; equivalent to x-intercepts.
X-Intercept: The specific point(s) where the graph crosses the x-axis.
Y-Intercept: The point(s) where the graph intersects the y-axis.

Function Behavior

Positive: Regions of the graph lying above the x-axis indicate positive function values.
Negative: Regions below the x-axis reflect negative function values.
Increasing: Describes the trend where, as input values (x) increase, the function values (f(x)) also increase.
Decreasing: Indicates that function values decrease as the input values increase.

Long-Term Behavior

End Behavior: Describes the behavior of the function values as x approaches positive or negative infinity.
Continuity: Refers to functions that are connected with no breaks, resulting in smooth curves or lines.
Extrema: The maximum and minimum values of a function, outlining the highest and lowest points on a graph.
Symmetry: Refers to functions exhibiting a vertical line that divides the graph into two symmetrical halves.

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Description

Explore essential definitions related to functions and continuity with these flashcards. Each card succinctly defines important concepts such as domain, range, and types of functions, facilitating a quick review for students. Ideal for anyone preparing for algebra exams or seeking to reinforce their understanding of mathematical functions.