Domain and Range in Functions

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Questions and Answers

What does set builder notation primarily represent?

  • The conditions that define the elements of a set (correct)
  • The exact numerical values of a domain or range
  • The graphical representation of a function
  • The sequential order of data in a table

When determining the least x-value for domain representation, which aspect is most crucial?

  • The highest point on the graph along the y-axis
  • The value where the graph begins on the x-axis (correct)
  • The average of all x-values present in the data
  • The slope of the graph at its steepest point

In set builder notation, how is the inclusion of a value expressed?

  • By omitting the value entirely
  • By stating the value is less than or equal to another
  • Using parentheses around the value
  • Using square brackets around the value (correct)

Which of the following best describes the term 'range' in relation to a graph?

<p>The interval of values along the y-axis (D)</p> Signup and view all the answers

What is the primary purpose of scanning from bottom to top along the y-axis when determining the range?

<p>To locate the minimum and maximum y-values in order (C)</p> Signup and view all the answers

What does the domain of a function represent?

<p>The set of all input values. (B)</p> Signup and view all the answers

Which of the following describes a continuous graph?

<p>Can be drawn without lifting the pencil. (A)</p> Signup and view all the answers

How is the range of a function best described?

<p>The set of all output values. (A)</p> Signup and view all the answers

In which form is the domain of a discrete function typically expressed?

<p>As a list of distinct points. (C)</p> Signup and view all the answers

What would a graph with no restrictions on values likely indicate about its domain?

<p>It represents all real numbers. (A)</p> Signup and view all the answers

When determining the domain of a continuous graph, what axis should you scan?

<p>X-axis. (C)</p> Signup and view all the answers

Which statement about set builder notation is correct for the range of a function?

<p>It can represent infinite points. (A)</p> Signup and view all the answers

What does it mean if a set of ordered pairs does not represent a function?

<p>At least one y-value is repeated for different x-values. (A)</p> Signup and view all the answers

Which symbol is used in set builder notation to indicate a closed interval where the endpoint is included?

<p>[ ] (A)</p> Signup and view all the answers

How would you represent the domain of the function if it includes the endpoints in set builder notation?

<p>{ x | a ≤ x ≤ b } (D)</p> Signup and view all the answers

Given a graph where the least x-value is included and the greatest x-value is not included, which set builder notation is correct?

<p>{ x | 2 ≤ x &lt; 5 } (B)</p> Signup and view all the answers

What is the purpose of using interval or set builder notation for domains and ranges?

<p>To clearly convey included and excluded values (B)</p> Signup and view all the answers

If a graph has an open circle at a x-value, how does it affect the set builder notation for the domain?

<p>The x-value should use &lt; in the notation (C)</p> Signup and view all the answers

What symbol represents an open interval in set builder notation?

<p>( ) (B)</p> Signup and view all the answers

Which of the following correctly represents a situation where the least y-value is not included and the greatest y-value is included?

<p>{ y | 1 &lt; y ≤ 10 } (C)</p> Signup and view all the answers

When using interval notation for a domain that extends indefinitely towards negative infinity and includes -3, what would be its representation?

<p>(-∞, -3] (D)</p> Signup and view all the answers

Flashcards

Domain of a function

The set of all possible input values (x-values) that can be plugged into a function.

Range of a function

The set of all possible output values (y-values) that a function can produce.

Function

A function is a set of ordered pairs where no two different ordered pairs have the same first element.

Vertical Line Test

A vertical line test is a way to determine if a graph represents a function by checking if any vertical line intersects the graph at more than one point.

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Table as a Function

To determine if a table represents a function, check if any x-value has more than one corresponding y-value.

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Domain

The set of allowable input values for a function, often represented by the x-axis.

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Range

The set of all possible output values for a function, often represented by the y-axis.

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Set-builder Notation

A method for representing a set of values using mathematical notation, describing values that satisfy a given condition.

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Set Builder Notation for Domain and Range

A notation where the domain is represented by the variable 'x' and the range is represented by the variable 'y'.

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Closed Endpoint in Set Builder Notation

In set builder notation for domain, brackets '[' and ']' indicate that the endpoint is included in the set, while parentheses '(' and ')' indicate the endpoint is excluded.

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Open Endpoint in Set Builder Notation

In set builder notation for domain, parentheses '(' and ')' indicate that the endpoint is excluded from the set, while brackets '[' and ']' indicate the endpoint is included.

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Least x-value

The smallest x-value of a graph can be used to determine the leftmost boundary of the domain.

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Greatest x-value

The largest x-value of a graph can be used to determine the rightmost boundary of the domain.

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What is the Domain?

The set of all possible input values (x-values) of a function.

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What is the Range?

The set of all possible output values (y-values) of a function.

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How do we write Domain and Range?

The way we write the domain and range depends on whether the graph is discrete (finite domain and range) or continuous (infinite domain and range). Discrete graphs have separate, distinct points and can be written as a list of values. Continuous graphs have connected points and are written as inequalities.

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What is the independent variable?

The independent variable is the input of a function, often represented by 'x'. It's the value that is changed or manipulated.

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What is the dependent variable?

The dependent variable is the output of a function, often represented by 'y'. Its value depends on the independent variable.

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How to determine the domain of a continuous graph?

To determine the domain of a continuous graph, scan the x-axis from left to right, noting any restrictions or breaks in the graph.

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How to determine the range of a continuous graph?

To determine the range of a continuous graph, scan the y-axis from bottom to top, noting any restrictions or breaks in the graph.

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What is a function?

A function is a relationship where each input (x-value) has exactly one output (y-value).

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Study Notes

Domain and Range Notes

  • Domain: The set of all input values (x-values) for a function. These values correspond to the points on the graph's x-axis.

  • Range: The set of all output values (y-values) for a function. These values correspond to the points on the graph's y-axis.

Writing Domain and Range

  • Discrete (Finite) Graphs: Domain and range consist of separate, distinct values. These are often represented by lists of numbers or points, like ordered pairs, tables, mappings, or scatter plots.

  • Continuous Graphs: The domain and range can be infinite and are often expressed as intervals or set-builder notation. A continuous graph is one that you can draw without lifting your pencil. These graphs can include lines, curves, or combinations of those.

Set Builder Notation

  • Use symbols like "x ∈" (x is an element of), "such that" or "≥" (greater than or equal to), "<" (less than), ">=" (greater than or equal to), and "<=" (less than or equal to) to describe the conditions for the domain and/or range.

Interval Notation

  • Uses parentheses and brackets to show whether or not endpoints are included in the interval. Parentheses exclude the endpoint (e.g., (2, 5) means the values between 2 and 5 but not including 2 or 5 themselves). Brackets include the endpoint (e.g., [2, 5] means all the values between 2 and 5, including 2 and 5).

Graphing to Find Domain and Range

  • Continuous graphs: Use the x-axis to determine the domain (scanning from left to right). Use the y-axis to determine the range (scanning from bottom to top).
  • Discrete graphs: Identify all the separate points on the graph. The x-coordinates represent the domain, and the y-coordinates represent the range.

Examples (from document)

  • Examples of discrete and continuous graphs, tables, mappings and ordered pairs show different ways to represent domains and ranges.

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Related Documents

22-23 Domain and Range Notes PDF

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