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What is a function?
What is a function?
A relation in which for every input there is exactly one output (for every x there is just one y).
What does the term 'input' refer to in a relation?
What does the term 'input' refer to in a relation?
The first coordinate of an ordered pair in a relation (x,y); also called the domain.
What does the term 'output' refer to in a relation?
What does the term 'output' refer to in a relation?
The second coordinate of an ordered pair in a relation (x,y); also called the range.
Is this relation a function? (-2,-2) (2,-1) (3,0) (5,3) (5,4)
Is this relation a function? (-2,-2) (2,-1) (3,0) (5,3) (5,4)
Is this relation a function? (-3,-1) (-2,0) (-1,3) (-1,0) (0,-1)
Is this relation a function? (-3,-1) (-2,0) (-1,3) (-1,0) (0,-1)
Is this relation a function? (4,11) (5,18) (6,25) (7,32) (8,39)
Is this relation a function? (4,11) (5,18) (6,25) (7,32) (8,39)
Is this relation a function? (10,1) (11,6) (12,11) (13,16)(14,21)
Is this relation a function? (10,1) (11,6) (12,11) (13,16)(14,21)
Is this relation a function? (0,1) (1,2) (2,3) (3,4) (4,4)
Is this relation a function? (0,1) (1,2) (2,3) (3,4) (4,4)
Function - passes the vertical line test. Is this a function?
Function - passes the vertical line test. Is this a function?
Function - each input has exactly one output. Is this a function?
Function - each input has exactly one output. Is this a function?
Not a function - does not pass the vertical line test. Is this a function?
Not a function - does not pass the vertical line test. Is this a function?
Not a function - there is an input (4) that goes to two different outputs (5 and -5). Is this a function?
Not a function - there is an input (4) that goes to two different outputs (5 and -5). Is this a function?
What is the vertical line test?
What is the vertical line test?
Describe a function in simple terms.
Describe a function in simple terms.
Provide an example of a function with ordered pairs.
Provide an example of a function with ordered pairs.
Provide an example of a function where inputs have the same output.
Provide an example of a function where inputs have the same output.
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Study Notes
Function Definition
- A function is a relation where each input (x) corresponds to exactly one output (y).
- Functions can be identified through ordered pairs, denoted as (x, y).
Key Terms
- Input: The first coordinate in an ordered pair; also known as the domain.
- Output: The second coordinate in an ordered pair; also known as the range.
Identifying Functions
- A relation is not a function if an input corresponds to multiple outputs. This is evident in examples with repeated first coordinates resulting in different second coordinates.
- Examples of relations that are not functions:
- (-2,-2), (2,-1), (3,0), (5,3), (5,4) fails because (5,3) and (5,4) have the same input.
- (-3,-1), (-2,0), (-1,3), (-1,0), (0,-1) fails because (-1,3) and (-1,0) have the same input.
Examples of Functions
- A relation is a function when every input has one unique output:
- (4,11), (5,18), (6,25), (7,32), (8,39) is a function due to unique outputs for each input.
- (0,1), (1,2), (2,3), (3,4), (4,4) meets the function criteria.
Vertical Line Test
- A graph represents a function if a vertical line intersects the graph at no more than one point.
- If a vertical line crosses at multiple points, the relation is classified as not a function.
Summary of Function Characteristics
- Functions consistently maintain one output for each input.
- Any instance of a shared input leading to multiple outputs rules out the possibility of a function designation.
Additional Notes
- Relations may still be called functions if mixed outputs occur only under distinct inputs.
- Examples of relations might vary but should adhere to the fundamental rules of input-output pairing to be considered functions.
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