Relating Domain to Context and Graphs - Algebra 2
Card 1 of 30
Identify the domain restriction for a square root $\sqrt{x-2}$.
Identify the domain restriction for a square root $\sqrt{x-2}$.
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Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.
Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.
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What is the domain of $f(x)=\sqrt{5-x}$?
What is the domain of $f(x)=\sqrt{5-x}$?
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$x\le 5$, so $(-\infty,5]$. The radicand $(5-x)$ must be non-negative.
$x\le 5$, so $(-\infty,5]$. The radicand $(5-x)$ must be non-negative.
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What is the domain of a function in terms of allowable inputs?
What is the domain of a function in terms of allowable inputs?
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The set of all allowable $x$-values (inputs). The domain defines which $x$-values are valid inputs for the function.
The set of all allowable $x$-values (inputs). The domain defines which $x$-values are valid inputs for the function.
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What does it mean if a graph has a point at $x=3$?
What does it mean if a graph has a point at $x=3$?
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$3$ is included in the domain. A plotted point shows the function is defined at that $x$-value.
$3$ is included in the domain. A plotted point shows the function is defined at that $x$-value.
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What does an open circle at $x=3$ on a graph indicate about the domain?
What does an open circle at $x=3$ on a graph indicate about the domain?
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$3$ is excluded from the domain. Open circles indicate undefined points, creating domain restrictions.
$3$ is excluded from the domain. Open circles indicate undefined points, creating domain restrictions.
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Identify the domain restriction caused by a denominator of $x-5$.
Identify the domain restriction caused by a denominator of $x-5$.
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Exclude $x=5$ from the domain. Denominators equal zero create undefined points.
Exclude $x=5$ from the domain. Denominators equal zero create undefined points.
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Identify the domain restriction for a square root $\sqrt{x-2}$.
Identify the domain restriction for a square root $\sqrt{x-2}$.
Tap to reveal answer
Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.
Require $x-2\ge 0$, so $x\ge 2$. Even roots require non-negative expressions under the radical.
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Identify the domain restriction for an even root $\sqrt{3-2x}$.
Identify the domain restriction for an even root $\sqrt{3-2x}$.
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Require $3-2x\ge 0$, so $x\le \frac{3}{2}$. Even roots need non-negative radicands, so solve the inequality.
Require $3-2x\ge 0$, so $x\le \frac{3}{2}$. Even roots need non-negative radicands, so solve the inequality.
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What is the typical domain of a polynomial function $f(x)$?
What is the typical domain of a polynomial function $f(x)$?
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All real numbers, $(-\infty,\infty)$. Polynomials are defined for all real number inputs.
All real numbers, $(-\infty,\infty)$. Polynomials are defined for all real number inputs.
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What is the typical domain of an exponential function $f(x)=a\cdot b^x$?
What is the typical domain of an exponential function $f(x)=a\cdot b^x$?
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All real numbers, $(-\infty,\infty)$. Exponential functions accept any real number exponent.
All real numbers, $(-\infty,\infty)$. Exponential functions accept any real number exponent.
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What is the typical domain of a logarithmic function $f(x)=\log(x)$?
What is the typical domain of a logarithmic function $f(x)=\log(x)$?
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Require $x>0$. Logarithms require positive arguments to be defined.
Require $x>0$. Logarithms require positive arguments to be defined.
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What domain restriction is created by $f(x)=\log(x-4)$?
What domain restriction is created by $f(x)=\log(x-4)$?
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Require $x-4>0$, so $x>4$. The logarithm argument $(x-4)$ must be positive.
Require $x-4>0$, so $x>4$. The logarithm argument $(x-4)$ must be positive.
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What is the domain of $f(x)=\frac{1}{x^2+1}$?
What is the domain of $f(x)=\frac{1}{x^2+1}$?
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All real numbers, $(-\infty,\infty)$. The denominator $x^2+1$ is never zero for real $x$.
All real numbers, $(-\infty,\infty)$. The denominator $x^2+1$ is never zero for real $x$.
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What is the domain of $f(x)=\frac{1}{x-7}$?
What is the domain of $f(x)=\frac{1}{x-7}$?
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All real numbers except $x=7$. The denominator equals zero when $x=7$, creating a restriction.
All real numbers except $x=7$. The denominator equals zero when $x=7$, creating a restriction.
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What is the domain of $f(x)=\sqrt{x}$?
What is the domain of $f(x)=\sqrt{x}$?
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$x\ge 0$, so $[0,\infty)$. Square roots require non-negative radicands.
$x\ge 0$, so $[0,\infty)$. Square roots require non-negative radicands.
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What is the domain of $f(x)=\log(x+2)$?
What is the domain of $f(x)=\log(x+2)$?
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$x>-2$, so $(-2,\infty)$. The logarithm argument $(x+2)$ must be positive.
$x>-2$, so $(-2,\infty)$. The logarithm argument $(x+2)$ must be positive.
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What is the domain of $f(x)=\frac{x+1}{x^2-9}$?
What is the domain of $f(x)=\frac{x+1}{x^2-9}$?
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All real numbers except $x=\pm 3$. Factor $x^2-9=(x-3)(x+3)$ to find where denominator equals zero.
All real numbers except $x=\pm 3$. Factor $x^2-9=(x-3)(x+3)$ to find where denominator equals zero.
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What is the domain of $f(x)=\frac{1}{(x-2)(x+5)}$?
What is the domain of $f(x)=\frac{1}{(x-2)(x+5)}$?
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All real numbers except $x=2$ and $x=-5$. Each factor in the denominator creates a domain restriction.
All real numbers except $x=2$ and $x=-5$. Each factor in the denominator creates a domain restriction.
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What is the domain of $f(x)=\frac{1}{\sqrt{x-3}}$?
What is the domain of $f(x)=\frac{1}{\sqrt{x-3}}$?
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Require $x-3>0$, so $(3,\infty)$. Square root in denominator requires strictly positive radicand.
Require $x-3>0$, so $(3,\infty)$. Square root in denominator requires strictly positive radicand.
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What is the domain of $f(x)=\log(2-x)$?
What is the domain of $f(x)=\log(2-x)$?
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Require $2-x>0$, so $(-\infty,2)$. Logarithm argument $(2-x)$ must be positive.
Require $2-x>0$, so $(-\infty,2)$. Logarithm argument $(2-x)$ must be positive.
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What is the domain of $f(x)=\frac{\sqrt{x-4}}{x-4}$?
What is the domain of $f(x)=\frac{\sqrt{x-4}}{x-4}$?
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Require $x>4$, so $(4,\infty)$. Both numerator and denominator require $x>4$.
Require $x>4$, so $(4,\infty)$. Both numerator and denominator require $x>4$.
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What is the domain of $f(x)=\sqrt{x^2-9}$?
What is the domain of $f(x)=\sqrt{x^2-9}$?
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$x\le -3$ or $x\ge 3$. Solve $x^2-9\ge 0$ using factoring and sign analysis.
$x\le -3$ or $x\ge 3$. Solve $x^2-9\ge 0$ using factoring and sign analysis.
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What is the domain of $f(x)=\frac{1}{x^2-4x+4}$?
What is the domain of $f(x)=\frac{1}{x^2-4x+4}$?
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All real numbers except $x=2$. The denominator $(x-2)^2$ equals zero only when $x=2$.
All real numbers except $x=2$. The denominator $(x-2)^2$ equals zero only when $x=2$.
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What is the domain of $f(x)=|x-5|$?
What is the domain of $f(x)=|x-5|$?
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All real numbers, $(-\infty,\infty)$. Absolute value functions are defined for all real numbers.
All real numbers, $(-\infty,\infty)$. Absolute value functions are defined for all real numbers.
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Which domain is appropriate for $h(n)$ = engines assembled when $n$ is a count?
Which domain is appropriate for $h(n)$ = engines assembled when $n$ is a count?
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Positive integers, $n\in{1,2,3,\dots}$. Engine counts must be positive whole numbers.
Positive integers, $n\in{1,2,3,\dots}$. Engine counts must be positive whole numbers.
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Which domain is appropriate for $A(r)=\pi r^2$ when $r$ is a radius?
Which domain is appropriate for $A(r)=\pi r^2$ when $r$ is a radius?
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Nonnegative real numbers, $r\ge 0$. Radius measurements cannot be negative in real contexts.
Nonnegative real numbers, $r\ge 0$. Radius measurements cannot be negative in real contexts.
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Which domain is appropriate for $t$ in a model describing time after a start?
Which domain is appropriate for $t$ in a model describing time after a start?
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Nonnegative real numbers, $t\ge 0$. Time after a starting point cannot be negative.
Nonnegative real numbers, $t\ge 0$. Time after a starting point cannot be negative.
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Which domain is appropriate for $d(t)$ distance traveled after $t$ seconds?
Which domain is appropriate for $d(t)$ distance traveled after $t$ seconds?
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Nonnegative real numbers, $t\ge 0$. Distance traveled cannot be negative in standard contexts.
Nonnegative real numbers, $t\ge 0$. Distance traveled cannot be negative in standard contexts.
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Which domain is appropriate for $P(n)$ population after $n$ years (counted in years)?
Which domain is appropriate for $P(n)$ population after $n$ years (counted in years)?
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Nonnegative integers, $n\in{0,1,2,\dots}$. Years counted as whole number increments from zero.
Nonnegative integers, $n\in{0,1,2,\dots}$. Years counted as whole number increments from zero.
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Which domain is appropriate for $C(x)$ cost to buy $x$ pounds of fruit?
Which domain is appropriate for $C(x)$ cost to buy $x$ pounds of fruit?
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Nonnegative real numbers, $x\ge 0$. Weight purchases cannot be negative quantities.
Nonnegative real numbers, $x\ge 0$. Weight purchases cannot be negative quantities.
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