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Questions and Answers
Which of the following equations represents a function?
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.
The vertical line test can be used to determine if a graph represents a function.
True (A)
Evaluate the function $f(x) = 2x + 7$ for $x = -5$. What is the result?
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 ______.
The output of the function $h(n) = -2n^2 + 4$ when $n = -4$ is ______.
Match the following functions with their evaluations:
Match the following functions with their evaluations:
Which of the following sets of ordered pairs represents a function?
Which of the following sets of ordered pairs represents a function?
The relation {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} is a function.
The relation {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} is a function.
What is the definition of a function?
What is the definition of a function?
In set notation, a relation is a function if each input has exactly one ________.
In set notation, a relation is a function if each input has exactly one ________.
Match the types of correspondence with their definitions:
Match the types of correspondence with their definitions:
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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) )
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