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Questions and Answers
A water dispenser usually has two buttons (red and blue) indicating whether you would want hot or cold water. Pressing any of these ______ will cause the dispenser to fill your container with hot or cold water.
A water dispenser usually has two buttons (red and blue) indicating whether you would want hot or cold water. Pressing any of these ______ will cause the dispenser to fill your container with hot or cold water.
buttons
In mathematics, a ______ is a kind of relation in which no two distinct ordered pairs have the same element.
In mathematics, a ______ is a kind of relation in which no two distinct ordered pairs have the same element.
function
In the function machine example with the electric fan, the fan blades spinning is the ______.
In the function machine example with the electric fan, the fan blades spinning is the ______.
output
In evaluating functions, simply replace all the x variables with whatever x has been ______.
In evaluating functions, simply replace all the x variables with whatever x has been ______.
If each value of the independent variable x is unique and associated with a unique value of the dependent variable y, the relation has a one-to-______ correspondence.
If each value of the independent variable x is unique and associated with a unique value of the dependent variable y, the relation has a one-to-______ correspondence.
Per every x variable, there is a corresponding y variable where x or independent variable is an ______, and y variable is a dependent variable or an output.
Per every x variable, there is a corresponding y variable where x or independent variable is an ______, and y variable is a dependent variable or an output.
Based in Oxford Languages a ______ is an activity or natural purpose to or intended for a person or a thing.
Based in Oxford Languages a ______ is an activity or natural purpose to or intended for a person or a thing.
If some values of both x and y are associated with more than one value of their counterpart, the relation has a many-to-______ correspondence.
If some values of both x and y are associated with more than one value of their counterpart, the relation has a many-to-______ correspondence.
In the function $y = 2x$, If you INPUT each value in the x variable you will have the y value as an ______ in the equation.
In the function $y = 2x$, If you INPUT each value in the x variable you will have the y value as an ______ in the equation.
Consider the table of values to determine the input, function, and the output. If the function is $y = 3x + 1$ where x is 2, the output is ______.
Consider the table of values to determine the input, function, and the output. If the function is $y = 3x + 1$ where x is 2, the output is ______.
In the function machine example of the electric fan, the ______ is the object where the button will act.
In the function machine example of the electric fan, the ______ is the object where the button will act.
When evaluating the function $f(x) = x + 3$ when x=0, the answer is ______.
When evaluating the function $f(x) = x + 3$ when x=0, the answer is ______.
When evaluating the piecewise function
$f(x) = x^2 - 6$ if $x = 1$,
the answer is ______.
When evaluating the piecewise function $f(x) = x^2 - 6$ if $x = 1$, the answer is ______.
If some values of x are associated with more than one value of y, the relation has a one-to-______ correspondence.
If some values of x are associated with more than one value of y, the relation has a one-to-______ correspondence.
Consider the function $f(x) = 3x - 4$. To find $f(2)$, you substitute 2 for ______ in the expression.
Consider the function $f(x) = 3x - 4$. To find $f(2)$, you substitute 2 for ______ in the expression.
In a water dispenser, the process of filling your container with hot or cold water is the ______.
In a water dispenser, the process of filling your container with hot or cold water is the ______.
With many-to-one correspondence, two or more values of ______ are associated with the same value of y.
With many-to-one correspondence, two or more values of ______ are associated with the same value of y.
The main concept or purpose for what something is made for is its ______.
The main concept or purpose for what something is made for is its ______.
In the example function $y = 2x$, each input is ______ after 'going through' the function.
In the example function $y = 2x$, each input is ______ after 'going through' the function.
Not all equations show ______ or contain only one output per every output.
Not all equations show ______ or contain only one output per every output.
Flashcards
What is a function?
What is a function?
An activity natural to its purpose, or intended for something.
What is an input?
What is an input?
The x variable; where per every x there is a corresponding y.
What is an output?
What is an output?
The y variable; where per every x there is a corresponding y.
One-to-one correspondence
One-to-one correspondence
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Many-to-one correspondence
Many-to-one correspondence
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One-to-many correspondence
One-to-many correspondence
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Many-to-many correspondence
Many-to-many correspondence
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Evaluating Functions
Evaluating Functions
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Study Notes
Introduction to Functions
- Functions are like everyday objects or people, each with a specific purpose.
- Oxford Languages defines a function as an activity with a natural purpose or intent for someone or something.
- In mathematics, a function is a type of relation where no two distinct ordered pairs share the same first element.
Exploring Functions
- For every x variable, there is a corresponding y variable.
- The x variable, also known as the independent variable, serves as the input.
- The y variable, or dependent variable, represents the output.
Electric Fan Example
- Input: Pressing the button on the fan.
- Function: The internal mechanism of the fan.
- Output: The fan blades spinning.
Example Function: y = 2x
- For the equation y = 2x e.g. if x = -2, then y = -4
Water Dispenser Example
- Input: Pressing the hot or cold water button.
- Function: The process of the dispenser filling the container.
- Output: Hot or cold water.
Table of Values Example
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | -5 | -2 | 0 | 4 | 7 |
- The x-values (-2, -1, 0, 1, 2) are the input.
- The y-values (-5, -2, 0, 4, 7) are the output.
- The function is y = 3x + 1, where each input value is tripled, and the result is increased by 1.
Relations
- Relations do not show functions, and may contain more than one output per every input.
One-to-One Correspondence
- Each value of the independent variable (x) is unique.
- Each x value is associated with a unique value of the dependent variable (y).
Many-to-One Correspondence
- Two or more values of x are associated with the same value of y.
One-to-Many Correspondence
- Some values of x are associated with more than one value of y.
Many-to-Many Correspondence
- Some values of both x and y are associated with more than one value of their counterpart.
Evaluating Functions
- Evaluating functions involves finding the value of f(x) or y for a given value of x.
- The approach is to replace all x variables with the assigned value.
Example 1: Evaluate f(x) = 3x - 4
- To evaluate at x = 2, substitute x = 2 into the function: f(2) = 3(2) - 4, which simplifies to f(2) = 2.
- To evaluate at x = 4, substitute x = 4 into the function: f(4) = 3(4) - 4, which simplifies to f(4) = 8.
Example 2: Piecewise Function
- Evaluate f(x) when (a) x = 0, (b) x = 1, and (c) x = 3.
- For x = 0, use the first expression: f(0) = 0 + 3 = 3.
- For x = 1, use the second expression: f(1) = (1)² - 6 = -5.
- For x = 3, use the third expression.
Graphing Functions
- The process is as follows:
- Tabulate arbitrary values of x.
- Obtain the corresponding values of f(x).
- Plot the ordered pairs on the Cartesian plane.
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