Understanding Functions, Domain, and Range - Algebra
Card 1 of 30
Evaluate $f(2)$ for $f(x)=3x-1$.
Evaluate $f(2)$ for $f(x)=3x-1$.
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$5$. Substitute $x=2$: $f(2)=3(2)-1=6-1=5$.
$5$. Substitute $x=2$: $f(2)=3(2)-1=6-1=5$.
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Find and correct the statement: “The graph of $f$ is the graph of $x=f(y)$.”
Find and correct the statement: “The graph of $f$ is the graph of $x=f(y)$.”
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Correct: The graph of $f$ is the graph of $y=f(x)$. The corrected notation shows proper function graphing.
Correct: The graph of $f$ is the graph of $y=f(x)$. The corrected notation shows proper function graphing.
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Identify the domain of the set ${(-2,5),(0,1),(4,1)}$.
Identify the domain of the set ${(-2,5),(0,1),(4,1)}$.
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${-2,0,4}$. Domain includes all unique first coordinates.
${-2,0,4}$. Domain includes all unique first coordinates.
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Identify the input-output pair for $f(x)=x-6$ when the input is $x=6$.
Identify the input-output pair for $f(x)=x-6$ when the input is $x=6$.
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$(6,0)$. Calculate $f(6)=6-6=0$, giving pair $(6,0)$.
$(6,0)$. Calculate $f(6)=6-6=0$, giving pair $(6,0)$.
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What is the correct reading of $f(7)=2$ in words?
What is the correct reading of $f(7)=2$ in words?
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The function value at $7$ is $2$. This describes evaluating the function at input $7$.
The function value at $7$ is $2$. This describes evaluating the function at input $7$.
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What is the correct reading of $f(x)$ in words?
What is the correct reading of $f(x)$ in words?
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Function of $x$. This is the standard way to read function notation.
Function of $x$. This is the standard way to read function notation.
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Identify whether ${(2,1),(3,1),(4,1),(5,1)}$ is a function.
Identify whether ${(2,1),(3,1),(4,1),(5,1)}$ is a function.
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Function. Each input maps to exactly one output value.
Function. Each input maps to exactly one output value.
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Identify whether ${(2,1),(2,1),(3,4)}$ is a function.
Identify whether ${(2,1),(2,1),(3,4)}$ is a function.
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Function (repeated pair does not create two outputs for one input). Repeated pairs don't violate the function definition.
Function (repeated pair does not create two outputs for one input). Repeated pairs don't violate the function definition.
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Find and correct the statement: “A function can assign two outputs to one input.”
Find and correct the statement: “A function can assign two outputs to one input.”
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Correct: A function assigns exactly one output to each input. The corrected statement reflects the function definition.
Correct: A function assigns exactly one output to each input. The corrected statement reflects the function definition.
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Identify the range of $f(x)=3$ over all real inputs.
Identify the range of $f(x)=3$ over all real inputs.
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${3}$. Constant functions have a single-element range.
${3}$. Constant functions have a single-element range.
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Identify the range of $f(x)=x+1$ over all real inputs.
Identify the range of $f(x)=x+1$ over all real inputs.
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All real numbers. Linear functions with non-zero slope have all reals as range.
All real numbers. Linear functions with non-zero slope have all reals as range.
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What must be true about the $y$-values for a single $x$ on the graph of a function?
What must be true about the $y$-values for a single $x$ on the graph of a function?
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There can be only one $y$-value for that $x$. Functions cannot have multiple $y$-values for one $x$-value.
There can be only one $y$-value for that $x$. Functions cannot have multiple $y$-values for one $x$-value.
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What test on a graph checks whether a relation is a function?
What test on a graph checks whether a relation is a function?
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The vertical line test. This graphical test checks the function definition property.
The vertical line test. This graphical test checks the function definition property.
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What is the output variable called when the function is written as $y=f(x)$?
What is the output variable called when the function is written as $y=f(x)$?
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$y$ (the dependent variable). The output depends on the input value chosen.
$y$ (the dependent variable). The output depends on the input value chosen.
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What is the input value called when the function is written as $y=f(x)$?
What is the input value called when the function is written as $y=f(x)$?
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$x$ (the independent variable). The input variable determines the output value.
$x$ (the independent variable). The input variable determines the output value.
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What is the output value called when the input is $x$ and the function is $f$?
What is the output value called when the input is $x$ and the function is $f$?
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$f(x)$. This is the standard notation for function output values.
$f(x)$. This is the standard notation for function output values.
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What is the key rule that distinguishes a function from a general relation?
What is the key rule that distinguishes a function from a general relation?
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No input is paired with more than one output. This is the fundamental property that defines a function.
No input is paired with more than one output. This is the fundamental property that defines a function.
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What ordered pair is plotted on the graph for an input $x$ of a function $f$?
What ordered pair is plotted on the graph for an input $x$ of a function $f$?
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$(x,f(x))$. The ordered pair shows input $x$ and corresponding output $f(x)$.
$(x,f(x))$. The ordered pair shows input $x$ and corresponding output $f(x)$.
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What is the meaning of the equation $y=f(x)$ in terms of a function’s graph?
What is the meaning of the equation $y=f(x)$ in terms of a function’s graph?
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It describes all points $(x,y)$ where $y$ equals the output $f(x)$. This creates a coordinate system where $x$ is input and $y$ is output.
It describes all points $(x,y)$ where $y$ equals the output $f(x)$. This creates a coordinate system where $x$ is input and $y$ is output.
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Choose the word that names the output set in a function definition: domain or range?
Choose the word that names the output set in a function definition: domain or range?
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Range. Range refers to the output set of a function.
Range. Range refers to the output set of a function.
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Choose the word that names the input set in a function definition: domain or range?
Choose the word that names the input set in a function definition: domain or range?
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Domain. Domain refers to the input set of a function.
Domain. Domain refers to the input set of a function.
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What does the statement “$x$ is an element of the domain” guarantee about $f(x)$?
What does the statement “$x$ is an element of the domain” guarantee about $f(x)$?
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$f(x)$ is defined and has exactly one value. Domain membership guarantees function value existence and uniqueness.
$f(x)$ is defined and has exactly one value. Domain membership guarantees function value existence and uniqueness.
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Identify whether the equation $y=x^3$ represents a function.
Identify whether the equation $y=x^3$ represents a function.
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Function. Each input gives exactly one output value.
Function. Each input gives exactly one output value.
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Identify whether the equation $y^2=x$ represents $y$ as a function of $x$.
Identify whether the equation $y^2=x$ represents $y$ as a function of $x$.
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Not a function (some $x$ values give two $y$ values). For example, when $x=4$, $y=±2$, giving two outputs.
Not a function (some $x$ values give two $y$ values). For example, when $x=4$, $y=±2$, giving two outputs.
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Identify whether the equation $y=3$ represents $y$ as a function of $x$.
Identify whether the equation $y=3$ represents $y$ as a function of $x$.
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Function (each $x$ gives the single output $3$). Horizontal lines pass the vertical line test.
Function (each $x$ gives the single output $3$). Horizontal lines pass the vertical line test.
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Identify whether the equation $x=3$ represents $y$ as a function of $x$.
Identify whether the equation $x=3$ represents $y$ as a function of $x$.
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Not a function (one $x$ value corresponds to many $y$ values). Vertical lines fail the vertical line test for functions.
Not a function (one $x$ value corresponds to many $y$ values). Vertical lines fail the vertical line test for functions.
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Identify the output when $f(x)=x^2-1$ and the input is $x=3$.
Identify the output when $f(x)=x^2-1$ and the input is $x=3$.
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$8$. Substitute $x=3$: $f(3)=3^2-1=9-1=8$.
$8$. Substitute $x=3$: $f(3)=3^2-1=9-1=8$.
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Identify the range of the set ${(1,0),(2,0),(3,4)}$.
Identify the range of the set ${(1,0),(2,0),(3,4)}$.
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${0,4}$. Range includes all unique second coordinates.
${0,4}$. Range includes all unique second coordinates.
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What is the range of $f(x)=x^2$ over the real numbers?
What is the range of $f(x)=x^2$ over the real numbers?
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All real numbers $y$ such that $y\ge 0$. Squaring any real number produces a non-negative result.
All real numbers $y$ such that $y\ge 0$. Squaring any real number produces a non-negative result.
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What is the domain of $f(x)=\sqrt{x}$ over the real numbers?
What is the domain of $f(x)=\sqrt{x}$ over the real numbers?
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All real numbers $x$ such that $x\ge 0$. Square roots require non-negative inputs in real numbers.
All real numbers $x$ such that $x\ge 0$. Square roots require non-negative inputs in real numbers.
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