Understand the Function Concept - 8th Grade Math
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What is the graph of a function, described using ordered pairs?
What is the graph of a function, described using ordered pairs?
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All ordered pairs $(x,y)$ where $y$ is the output for input $x$. The graph shows all input-output relationships.
All ordered pairs $(x,y)$ where $y$ is the output for input $x$. The graph shows all input-output relationships.
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What is an ordered pair and what do its coordinates represent?
What is an ordered pair and what do its coordinates represent?
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A point $(x,y)$ with input $x$ and output $y$. Written as $(input, output)$ format.
A point $(x,y)$ with input $x$ and output $y$. Written as $(input, output)$ format.
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What is the range of a relation written as ordered pairs?
What is the range of a relation written as ordered pairs?
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The set of all output values (all $y$-values). Range contains all possible outputs.
The set of all output values (all $y$-values). Range contains all possible outputs.
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What is the domain of a relation written as ordered pairs?
What is the domain of a relation written as ordered pairs?
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The set of all input values (all $x$-values). Domain contains all possible inputs.
The set of all input values (all $x$-values). Domain contains all possible inputs.
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What is a function in terms of inputs and outputs?
What is a function in terms of inputs and outputs?
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A rule that assigns each input exactly one output. This is the fundamental definition of a function.
A rule that assigns each input exactly one output. This is the fundamental definition of a function.
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What does the vertical line test say about functions?
What does the vertical line test say about functions?
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A graph is a function if no vertical line hits it more than once. Each vertical line represents one input value.
A graph is a function if no vertical line hits it more than once. Each vertical line represents one input value.
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What condition must be true for a relation to be a function?
What condition must be true for a relation to be a function?
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No input can be paired with more than one output. Each input must have a unique output.
No input can be paired with more than one output. Each input must have a unique output.
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Which test on a graph shows whether a relation is a function?
Which test on a graph shows whether a relation is a function?
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The vertical line test. A visual method to check if a relation is a function.
The vertical line test. A visual method to check if a relation is a function.
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Identify whether the equation $x=2$ defines $y$ as a function of $x$.
Identify whether the equation $x=2$ defines $y$ as a function of $x$.
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No; $x=2$ has many possible $y$-values. A vertical line has undefined slope and multiple $y$-values.
No; $x=2$ has many possible $y$-values. A vertical line has undefined slope and multiple $y$-values.
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Identify the domain of ${(-1,2),(0,4),(3,4)}$.
Identify the domain of ${(-1,2),(0,4),(3,4)}$.
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Domain $={-1,0,3}$. List all first coordinates from the ordered pairs.
Domain $={-1,0,3}$. List all first coordinates from the ordered pairs.
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Which statement best distinguishes a function from a general relation?
Which statement best distinguishes a function from a general relation?
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A function has exactly one output for each input. The key difference is the one-to-one input-output rule.
A function has exactly one output for each input. The key difference is the one-to-one input-output rule.
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Which graph description guarantees a relation is not a function?
Which graph description guarantees a relation is not a function?
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A vertical line intersects the graph at more than one point. This means one input has multiple outputs.
A vertical line intersects the graph at more than one point. This means one input has multiple outputs.
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Identify whether ${(0,5),(1,5),(2,5)}$ is a function.
Identify whether ${(0,5),(1,5),(2,5)}$ is a function.
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Function; each input has exactly one output. Multiple inputs can share the same output.
Function; each input has exactly one output. Multiple inputs can share the same output.
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Identify whether ${(1,2),(1,3),(2,4)}$ is a function.
Identify whether ${(1,2),(1,3),(2,4)}$ is a function.
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Not a function because input $1$ has two outputs. Input $1$ maps to both $2$ and $3$, violating the function rule.
Not a function because input $1$ has two outputs. Input $1$ maps to both $2$ and $3$, violating the function rule.
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Which ordered pair represents the output when the input is $x=4$?
Which ordered pair represents the output when the input is $x=4$?
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The point $(4,y)$ on the graph. Find where the graph crosses the vertical line $x=4$.
The point $(4,y)$ on the graph. Find where the graph crosses the vertical line $x=4$.
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Identify whether the equation $y=3x-2$ defines a function.
Identify whether the equation $y=3x-2$ defines a function.
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Yes; each $x$ gives exactly one $y$. For any $x$, there's only one $y$ value: $y=3x-2$.
Yes; each $x$ gives exactly one $y$. For any $x$, there's only one $y$ value: $y=3x-2$.
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Identify the range of ${(-1,2),(0,4),(3,4)}$.
Identify the range of ${(-1,2),(0,4),(3,4)}$.
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Range $={2,4}$. List unique second coordinates; $4$ appears twice but count once.
Range $={2,4}$. List unique second coordinates; $4$ appears twice but count once.
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Which variable is typically the input and which is the output in $y=f(x)$?
Which variable is typically the input and which is the output in $y=f(x)$?
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$x$ is the input; $y$ is the output. Standard notation: $f$ maps input $x$ to output $y$.
$x$ is the input; $y$ is the output. Standard notation: $f$ maps input $x$ to output $y$.
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What is the definition of a function in terms of inputs and outputs?
What is the definition of a function in terms of inputs and outputs?
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A rule that assigns each input exactly one output. This ensures each input has a unique, predictable output.
A rule that assigns each input exactly one output. This ensures each input has a unique, predictable output.
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What must be true about outputs if two ordered pairs share the same input in a function?
What must be true about outputs if two ordered pairs share the same input in a function?
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The outputs must be equal. A function cannot map one input to different outputs.
The outputs must be equal. A function cannot map one input to different outputs.
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What is the meaning of the ordered pair $(x,y)$ on a function’s graph?
What is the meaning of the ordered pair $(x,y)$ on a function’s graph?
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$x$ is the input and $y$ is the corresponding output. Ordered pairs show the input-output relationship.
$x$ is the input and $y$ is the corresponding output. Ordered pairs show the input-output relationship.
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What is the graph of a function described as, using ordered pairs?
What is the graph of a function described as, using ordered pairs?
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The set of all ordered pairs $(x,y)$ with $y=f(x)$. Each point represents an input-output pair.
The set of all ordered pairs $(x,y)$ with $y=f(x)$. Each point represents an input-output pair.
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Identify whether the relation ${(1,2),(1,3)}$ is a function.
Identify whether the relation ${(1,2),(1,3)}$ is a function.
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Not a function. Input $1$ maps to two different outputs ($2$ and $3$).
Not a function. Input $1$ maps to two different outputs ($2$ and $3$).
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Identify whether the relation ${(1,2),(2,2),(3,2)}$ is a function.
Identify whether the relation ${(1,2),(2,2),(3,2)}$ is a function.
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Function. Each input has exactly one output.
Function. Each input has exactly one output.
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What is the output when $f(x)=2x+1$ and the input is $x=3$?
What is the output when $f(x)=2x+1$ and the input is $x=3$?
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$7$. Substitute: $f(3) = 2(3) + 1 = 6 + 1$.
$7$. Substitute: $f(3) = 2(3) + 1 = 6 + 1$.
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What is the output when $g(x)=x^2$ and the input is $x=-4$?
What is the output when $g(x)=x^2$ and the input is $x=-4$?
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$16$. Substitute: $g(-4) = (-4)^2 = 16$.
$16$. Substitute: $g(-4) = (-4)^2 = 16$.
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What is $f(0)$ for the function $f(x)=-3x+5$?
What is $f(0)$ for the function $f(x)=-3x+5$?
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$5$. Substitute: $f(0) = -3(0) + 5 = 0 + 5$.
$5$. Substitute: $f(0) = -3(0) + 5 = 0 + 5$.
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Identify the input and output in the ordered pair $( -2, 7 )$.
Identify the input and output in the ordered pair $( -2, 7 )$.
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Input $-2$; output $7$. First coordinate is input, second is output.
Input $-2$; output $7$. First coordinate is input, second is output.
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Which test determines whether a graph represents a function by checking vertical lines?
Which test determines whether a graph represents a function by checking vertical lines?
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The vertical line test. If any vertical line hits twice, it's not a function.
The vertical line test. If any vertical line hits twice, it's not a function.
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What does it mean if a vertical line intersects a graph more than once?
What does it mean if a vertical line intersects a graph more than once?
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The relation is not a function. One input would have multiple outputs.
The relation is not a function. One input would have multiple outputs.
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