Algebra Class - Guess the Function

Questions and Answers

Which of the following equations represents a function?

y = x^2 - 1 (correct)

y = 3x + 2 (correct)

x = 5

y^2 = x + 1

The vertical line test can be used to determine if a graph represents a function.

True

Evaluate the function $f(x) = 2x + 7$ for $x = -5$. What is the result?

-3

The output of the function $h(n) = -2n^2 + 4$ when $n = -4$ is ______.

36 Signup and view all the answers

Match the following functions with their evaluations:

w(t) = -2t + 1 = w(7) = -13 h(n) = -2n^2 + 4 = h(-4) = -28 w(a) = a + 3 = w(6) = 9 Signup and view all the answers

Which of the following sets of ordered pairs represents a function?

{(2, 4), (3, 4), (4, 4), (5, 4), (6,4)} Signup and view all the answers

The relation {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} is a function.

False Signup and view all the answers

What is the definition of a function?

A relation where each input corresponds to exactly one output. Signup and view all the answers

In set notation, a relation is a function if each input has exactly one ________.

output Signup and view all the answers

Match the types of correspondence with their definitions:

One-to-one = Each input corresponds to exactly one unique output. Many-to-one = Multiple inputs correspond to the same output. One-to-many = One input corresponds to multiple outputs. Many-to-many = Multiple inputs correspond to multiple outputs. Signup and view all the answers

Study Notes

Functions and Relations

A function relates an input (x) to exactly one output (y).
Example of a function: ( y = 2x ) - values yield:
- ( x = 2 ) results in ( y = 4 )
- ( x = 5 ) results in ( y = 10 )
- ( x = -8 ) results in ( y = -16 )

Non-Functional Relations

A relation with the same x-value producing different y-values is not a function.
Example of non-function:
- ( x = 3 ) yields ( y = -1 ) and ( y = -5 )

Function Properties

One-to-one correspondence: Each input corresponds to one unique output.
Many-to-one correspondence: Multiple inputs can map to the same output, still considered a function.
One-to-many correspondence: Invalidates the relation as a function.

Set Notation and Function Evaluation

Example set that is a function:
- ( {(2, 3), (3, 0), (5, 2), (4, 3)} )
- Each x-value is unique.
Example set that is not a function:
- ( {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} )
- The x-value 5 maps to two different y-values (2 and 3).

Identifying Functions from Ordered Pairs

Set A: Not a function due to repeating x-values.
Set B: Not a function, all x-values are 3.
Set C: Function, repeated x-values map to the same y-value.
Set D: Not a function, x-value 4 maps to multiple y-values.

Graphical Representation

A graph represents a function if every vertical line crosses the graph at most once.
Using the vertical line test helps determine if a graph is a function.

Evaluating Functions

Substitute the x-value into the function to find the corresponding y-value.
Example evaluations for ( f(x) = 2x + 7 ):
- For ( x = -5 ): ( f(-5) = -3 )
- For ( x = -2 ): ( f(-2) = 3 )
- For ( x = 9 ): ( f(9) = 25 )

Assessment Tasks

Assess whether given values or sets define functions.
Evaluate specific functions with given x-values:
- Example function ( w(t) = -2t + 1 ) for ( w(7) )
- Example function ( h(n) = -2n^2 + 4 ) for ( h(-4) )
- Example function ( w(a) = a + 3 ) for ( w(6) )

Description

This quiz challenges you to determine the functions represented by given sets of input and output values. Analyze the relationships between x and y to identify whether they represent linear functions. Perfect for reinforcing your understanding of functional relationships in algebra.