Mathematics Relations and Functions

Questions and Answers

What type of mapping diagram is represented by a set of ordered pairs?

Function

Relation (correct)

Mapping

None of the above

What is the range Y of the relation defined by $x \rightarrow x + 3$ for the domain $X = {0, 1, 2, 3}$?

\{1, 2, 3, 4\}

\{0, 1, 2, 3\}

\{4, 5, 6, 7\}

\{3, 4, 5, 6\} (correct)

In the notation of functions, what does $f(x)$ typically represent?

The output value (correct)

The input value

The set of all ordered pairs

The domain

For the function $f: x \rightarrow 3x^2 + 2x - 1$, what is $f(3)$?

26 Signup and view all the answers

What is the image of x, denoted by $f(x)$?

$3x^2 + 2x - 1$ Signup and view all the answers

When completing the table of values for the function $f: x \rightarrow 2x^2$ over the domain $-4 \leq x \leq 4$, what is $f(-2)$?

8 Signup and view all the answers

What does the set of all y-values in a function describe?

Range Signup and view all the answers

Based on the function $f: x \rightarrow 3x^2 + 2x - 1$, which of the following describes the behavior of the function?

It is a quadratic function. Signup and view all the answers

Study Notes

Relations and Functions

A mapping diagram visually represents the relationship between a set of inputs (domain) and outputs (range).
A set of ordered pairs follows a specific rule, indicating how each input is associated with its output.
The set of all y-values in a function is called the range.

Key Concepts in Mapping Diagrams

Different types of mapping diagrams are used to show various functions and relationships.
The image of a specific input ( x ) is denoted by ( f(x) ), indicating the output value corresponding to that input.

Problem-Solving Techniques

To find the range ( Y ) for a relation like ( x \to x + 3 ) given the domain ( X = {0, 1, 2, 3} ), substitute each value of ( x ) into the function:
- For ( x = 0 ), ( Y = 0 + 3 = 3 )
- For ( x = 1 ), ( Y = 1 + 3 = 4 )
- For ( x = 2 ), ( Y = 2 + 3 = 5 )
- For ( x = 3 ), ( Y = 3 + 3 = 6 )
- Resulting range ( Y = {3, 4, 5, 6} )

Evaluating Functions

To evaluate the function ( f(x) = 3x^2 + 2x - 1 ) at ( x = 3 ):
- Substitute ( 3 ) into the function:
- ( f(3) = 3(3)^2 + 2(3) - 1 = 27 + 6 - 1 = 32 ).

Quadratic Functions

The quadratic function ( f(x) = 2x^2 ) requires completion of a table for the domain ( -4 \leq x \leq 4 ):
- Calculate ( f(x) ) for each ( x ) value:
  - ( f(-4) = 32 )
  - ( f(-3) = 18 )
  - ( f(-2) = 8 )
  - ( f(-1) = 2 )
  - ( f(0) = 0 )
  - ( f(1) = 2 )
  - ( f(2) = 8 )
  - ( f(3) = 18 )
  - ( f(4) = 32 )

Graphing Quadratic Functions

Graph the function using the calculated values in the table to visualize the parabolic shape of quadratic functions, ensuring accuracy by plotting key points along the x-axis from -4 to 4.

Description

This quiz covers the essential concepts of relations and functions, focusing on mapping diagrams and ordered pairs. You'll explore how each input relates to its corresponding output and determine the range of given functions. Test your understanding of these foundational mathematical principles.