Restrict Domain to Make Invertible - Algebra 2
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Which domain restriction makes $f(x)=x^2$ invertible and uses the negative branch?
Which domain restriction makes $f(x)=x^2$ invertible and uses the negative branch?
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Restrict to $x\le 0$. This restriction uses the left branch of the parabola.
Restrict to $x\le 0$. This restriction uses the left branch of the parabola.
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Which domain restriction makes $f(x)=x^2$ invertible and matches the principal square root?
Which domain restriction makes $f(x)=x^2$ invertible and matches the principal square root?
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Restrict to $x\ge 0$. This restriction uses the right branch of the parabola.
Restrict to $x\ge 0$. This restriction uses the right branch of the parabola.
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What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\ge -2$?
What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\ge -2$?
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$f^{-1}(x)=-2+\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the positive branch.
$f^{-1}(x)=-2+\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the positive branch.
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What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\le 3$?
What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\le 3$?
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$f^{-1}(x)=3-\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the negative square root.
$f^{-1}(x)=3-\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the negative square root.
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What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\ge 3$?
What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\ge 3$?
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$f^{-1}(x)=3+\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the positive square root.
$f^{-1}(x)=3+\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the positive square root.
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What restriction makes $f(x)=(x-3)^2$ invertible to match the negative square root branch?
What restriction makes $f(x)=(x-3)^2$ invertible to match the negative square root branch?
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Restrict to $x\le 3$. The vertex is at $x=3$, so restrict to the left side.
Restrict to $x\le 3$. The vertex is at $x=3$, so restrict to the left side.
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What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\le 0$?
What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\le 0$?
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$f^{-1}(x)=-\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the negative square root.
$f^{-1}(x)=-\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the negative square root.
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What is the inverse of $f(x)=x^2$ if the restricted domain is $[-2,0]$?
What is the inverse of $f(x)=x^2$ if the restricted domain is $[-2,0]$?
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$f^{-1}(x)=-\sqrt{x}$ for $0\le x\le 4$. The range $[0,4]$ comes from squaring the domain $[-2,0]$.
$f^{-1}(x)=-\sqrt{x}$ for $0\le x\le 4$. The range $[0,4]$ comes from squaring the domain $[-2,0]$.
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What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the left branch?
What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the left branch?
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Restrict to $x\le h$. The vertex $x=h$ is the axis of symmetry of the parabola.
Restrict to $x\le h$. The vertex $x=h$ is the axis of symmetry of the parabola.
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What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\le 3$?
What is $f^{-1}(x)$ for $f(x)=(x-3)^2$ with restricted domain $x\le 3$?
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$f^{-1}(x)=3-\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the negative square root.
$f^{-1}(x)=3-\sqrt{x}$. Solve $y=(x-3)^2$ for $x$ using the negative square root.
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What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\ge -2$?
What is $f^{-1}(x)$ for $f(x)=-(x+2)^2$ with restricted domain $x\ge -2$?
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$f^{-1}(x)=-2+\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the positive branch.
$f^{-1}(x)=-2+\sqrt{-x}$. Solve $y=-(x+2)^2$ for $x$ using the positive branch.
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What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the right branch?
What restriction makes $f(x)=(x-h)^2+k$ one-to-one using the right branch?
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Restrict to $x\ge h$. The vertex $x=h$ is the axis of symmetry of the parabola.
Restrict to $x\ge h$. The vertex $x=h$ is the axis of symmetry of the parabola.
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What restriction makes $f(x)=-(x+2)^2$ invertible using the right half of the parabola?
What restriction makes $f(x)=-(x+2)^2$ invertible using the right half of the parabola?
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Restrict to $x\ge -2$. The vertex is at $x=-2$, so restrict to the right side.
Restrict to $x\ge -2$. The vertex is at $x=-2$, so restrict to the right side.
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What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\le 0$?
What is $f^{-1}(x)$ for $f(x)=x^2-4$ with restricted domain $x\le 0$?
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$f^{-1}(x)=-\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the negative square root.
$f^{-1}(x)=-\sqrt{x+4}$. Solve $y=x^2-4$ for $x$ using the negative square root.
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Identify the restricted domain that makes $f(x)=x^2$ invertible if the domain is $[-2,2]$.
Identify the restricted domain that makes $f(x)=x^2$ invertible if the domain is $[-2,2]$.
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Restrict to $[0,2]$ or $[-2,0]$. Choose one monotonic piece from each side of the vertex at $x=0$.
Restrict to $[0,2]$ or $[-2,0]$. Choose one monotonic piece from each side of the vertex at $x=0$.
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What is the range of $f(x)=(x-3)^2$ when the domain is restricted to $x\ge 3$?
What is the range of $f(x)=(x-3)^2$ when the domain is restricted to $x\ge 3$?
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Range is $y\ge 0$. The parabola opens upward with vertex at $(3,0)$.
Range is $y\ge 0$. The parabola opens upward with vertex at $(3,0)$.
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What is the range of $f^{-1}(x)$ if $f$ is restricted to the domain $x\le 3$?
What is the range of $f^{-1}(x)$ if $f$ is restricted to the domain $x\le 3$?
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Range of $f^{-1}$ is $x\le 3$. The range of the inverse equals the domain of the original function.
Range of $f^{-1}$ is $x\le 3$. The range of the inverse equals the domain of the original function.
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What is the first algebraic step to find an inverse after restricting the domain?
What is the first algebraic step to find an inverse after restricting the domain?
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Write $y=f(x)$ and swap $x$ and $y$. This sets up the equation to solve for the inverse.
Write $y=f(x)$ and swap $x$ and $y$. This sets up the equation to solve for the inverse.
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What must you check after solving for $y$ when finding an inverse of a restricted quadratic?
What must you check after solving for $y$ when finding an inverse of a restricted quadratic?
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Choose the correct sign to match the restriction. The sign must correspond to the restricted domain interval.
Choose the correct sign to match the restriction. The sign must correspond to the restricted domain interval.
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Identify the correct inverse for restricted $f(x)=(x-4)^2$ with domain $x\ge 4$.
Identify the correct inverse for restricted $f(x)=(x-4)^2$ with domain $x\ge 4$.
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$f^{-1}(x)=4+\sqrt{x}$. For $x\ge 4$, use the positive square root.
$f^{-1}(x)=4+\sqrt{x}$. For $x\ge 4$, use the positive square root.
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Identify the restricted domain that makes $f(x)=(x+5)^2$ invertible using the left branch.
Identify the restricted domain that makes $f(x)=(x+5)^2$ invertible using the left branch.
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Restrict to $x\le -5$. The vertex is at $x=-5$, so restrict to the left side.
Restrict to $x\le -5$. The vertex is at $x=-5$, so restrict to the left side.
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Identify the restricted domain that makes $f(x)=-(x-1)^2+7$ invertible using the right branch.
Identify the restricted domain that makes $f(x)=-(x-1)^2+7$ invertible using the right branch.
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Restrict to $x\ge 1$. The vertex is at $x=1$, so restrict to the right side.
Restrict to $x\ge 1$. The vertex is at $x=1$, so restrict to the right side.
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What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\ge 1$?
What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\ge 1$?
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$f^{-1}(x)=1+\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the right branch.
$f^{-1}(x)=1+\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the right branch.
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What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\le 1$?
What is $f^{-1}(x)$ for $f(x)=-(x-1)^2+7$ with restricted domain $x\le 1$?
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$f^{-1}(x)=1-\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the left branch.
$f^{-1}(x)=1-\sqrt{7-x}$. Solve $y=-(x-1)^2+7$ for $x$ using the left branch.
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Which restriction makes $f(x)=x^2+2x$ invertible by using the increasing side of the parabola?
Which restriction makes $f(x)=x^2+2x$ invertible by using the increasing side of the parabola?
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Restrict to $x\ge -1$. Complete the square to find vertex at $x=-1$, then use right side.
Restrict to $x\ge -1$. Complete the square to find vertex at $x=-1$, then use right side.
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Which restriction makes $f(x)=x^2+2x$ invertible by using the decreasing side of the parabola?
Which restriction makes $f(x)=x^2+2x$ invertible by using the decreasing side of the parabola?
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Restrict to $x\le -1$. Complete the square to find vertex at $x=-1$, then use left side.
Restrict to $x\le -1$. Complete the square to find vertex at $x=-1$, then use left side.
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What happens to domain and range when taking an inverse $f^{-1}$?
What happens to domain and range when taking an inverse $f^{-1}$?
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Domain and range swap. The input and output sets exchange roles.
Domain and range swap. The input and output sets exchange roles.
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What is the relationship between the graphs of $f$ and $f^{-1}$?
What is the relationship between the graphs of $f$ and $f^{-1}$?
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They reflect across the line $y=x$. The graphs are mirror images across the diagonal line $y=x$.
They reflect across the line $y=x$. The graphs are mirror images across the diagonal line $y=x$.
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What is the goal of restricting a domain to make a function invertible?
What is the goal of restricting a domain to make a function invertible?
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Make the function one-to-one. Only one-to-one functions have inverses that are also functions.
Make the function one-to-one. Only one-to-one functions have inverses that are also functions.
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What does restricting the domain of a function mean?
What does restricting the domain of a function mean?
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Limit inputs to a subset of the original domain. This creates a new function with a smaller domain.
Limit inputs to a subset of the original domain. This creates a new function with a smaller domain.
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