Verify Functions Are Inverses - Algebra 2
Card 1 of 30
What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ without restricting $f$ to $x\ge 0$?
What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ without restricting $f$ to $x\ge 0$?
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$g(f(x))=|x|$. Square root returns absolute value for all real inputs.
$g(f(x))=|x|$. Square root returns absolute value for all real inputs.
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What is $g(f(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?
What is $g(f(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?
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$g(f(x))=x$. Substituting: $g(2x-5)=\frac{(2x-5)+5}{2}=x$.
$g(f(x))=x$. Substituting: $g(2x-5)=\frac{(2x-5)+5}{2}=x$.
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Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $f(g(x))$.
Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $f(g(x))$.
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$f(g(x))=x$. Substituting: $f(x-7)=(x-7)+7=x$.
$f(g(x))=x$. Substituting: $f(x-7)=(x-7)+7=x$.
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What is the inverse relationship between domain and range for inverse functions?
What is the inverse relationship between domain and range for inverse functions?
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$\text{Dom}(f)=\text{Ran}(g)$ and $\text{Ran}(f)=\text{Dom}(g)$. Domain and range swap between inverse functions.
$\text{Dom}(f)=\text{Ran}(g)$ and $\text{Ran}(f)=\text{Dom}(g)$. Domain and range swap between inverse functions.
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Identify $f(g(x))$ if $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x}$ with $x\ne 0$.
Identify $f(g(x))$ if $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x}$ with $x\ne 0$.
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$f(g(x))=x$ for $x\ne 0$. Substituting: $f(\frac{1}{x})=\frac{1}{\frac{1}{x}}=x$.
$f(g(x))=x$ for $x\ne 0$. Substituting: $f(\frac{1}{x})=\frac{1}{\frac{1}{x}}=x$.
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What coordinate swap describes the inverse relationship between points on $f$ and $f^{-1}$?
What coordinate swap describes the inverse relationship between points on $f$ and $f^{-1}$?
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If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$. Coordinates swap between function and inverse.
If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$. Coordinates swap between function and inverse.
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What is $g(f(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?
What is $g(f(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?
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$g(f(x))=x$. Substituting: $g(3x)=\frac{3x}{3}=x$.
$g(f(x))=x$. Substituting: $g(3x)=\frac{3x}{3}=x$.
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What must be checked about domains when verifying inverses by composition?
What must be checked about domains when verifying inverses by composition?
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Each composition must equal $x$ for all $x$ in its stated domain. Compositions must work on their proper domains.
Each composition must equal $x$ for all $x$ in its stated domain. Compositions must work on their proper domains.
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What test is commonly used to decide whether a function is one-to-one before finding an inverse?
What test is commonly used to decide whether a function is one-to-one before finding an inverse?
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The horizontal line test. Checks if every horizontal line intersects once.
The horizontal line test. Checks if every horizontal line intersects once.
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Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $g(f(x))$.
Identify the result of composing $f(x)=x+7$ and $g(x)=x-7$ as $g(f(x))$.
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$g(f(x))=x$. Substituting: $g(x+7)=(x+7)-7=x$.
$g(f(x))=x$. Substituting: $g(x+7)=(x+7)-7=x$.
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What is $g(f(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?
What is $g(f(x))$ if $f(x)=2x-5$ and $g(x)=\frac{x+5}{2}$?
Tap to reveal answer
$g(f(x))=x$. Substituting: $g(2x-5)=\frac{(2x-5)+5}{2}=x$.
$g(f(x))=x$. Substituting: $g(2x-5)=\frac{(2x-5)+5}{2}=x$.
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What is $g(f(x))$ if $f(x)=\frac{x-4}{5}$ and $g(x)=5x+4$?
What is $g(f(x))$ if $f(x)=\frac{x-4}{5}$ and $g(x)=5x+4$?
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$g(f(x))=x$. Substituting: $g(\frac{x-4}{5})=5\cdot\frac{x-4}{5}+4=x$.
$g(f(x))=x$. Substituting: $g(\frac{x-4}{5})=5\cdot\frac{x-4}{5}+4=x$.
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What is $g(f(x))$ if $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$?
What is $g(f(x))$ if $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$?
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$g(f(x))=x$. Substituting: $g(x^3)=\sqrt[3]{x^3}=x$.
$g(f(x))=x$. Substituting: $g(x^3)=\sqrt[3]{x^3}=x$.
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What is the key simplification goal when checking $f(g(x))$ to verify inverses?
What is the key simplification goal when checking $f(g(x))$ to verify inverses?
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Simplify until the result is exactly $x$. The composition should reduce to the identity.
Simplify until the result is exactly $x$. The composition should reduce to the identity.
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What is $g(f(x))$ if $f(x)=\frac{4}{x}$ and $g(x)=\frac{4}{x}$ with $x\ne 0$?
What is $g(f(x))$ if $f(x)=\frac{4}{x}$ and $g(x)=\frac{4}{x}$ with $x\ne 0$?
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$g(f(x))=x$ for $x\ne 0$. Substituting: $g(\frac{4}{x})=\frac{4}{\frac{4}{x}}=x$.
$g(f(x))=x$ for $x\ne 0$. Substituting: $g(\frac{4}{x})=\frac{4}{\frac{4}{x}}=x$.
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Identify $f(g(x))$ if $f(x)=\ln(x)$ and $g(x)=e^x$.
Identify $f(g(x))$ if $f(x)=\ln(x)$ and $g(x)=e^x$.
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$f(g(x))=x$. Natural log and exponential are inverses.
$f(g(x))=x$. Natural log and exponential are inverses.
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Identify $f(g(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
Identify $f(g(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
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$f(g(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.
$f(g(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.
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Identify $g(f(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
Identify $g(f(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
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$g(f(x))=x$. Exponential and natural log are inverses.
$g(f(x))=x$. Exponential and natural log are inverses.
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What is $f(g(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?
What is $f(g(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?
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$f(g(x))=x$ for $x>0$. Domain restriction needed for $\log_2(x)$.
$f(g(x))=x$ for $x>0$. Domain restriction needed for $\log_2(x)$.
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What is $g(f(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?
What is $g(f(x))$ if $f(x)=2^x$ and $g(x)=\log_2(x)$?
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$g(f(x))=x$. Exponential and logarithm base 2 are inverses.
$g(f(x))=x$. Exponential and logarithm base 2 are inverses.
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What is $f(g(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?
What is $f(g(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?
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$f(g(x))=x$. Substituting: $f(\sqrt[3]{x-1})=(\sqrt[3]{x-1})^3+1=x$.
$f(g(x))=x$. Substituting: $f(\sqrt[3]{x-1})=(\sqrt[3]{x-1})^3+1=x$.
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What is $g(f(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?
What is $g(f(x))$ if $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$?
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$g(f(x))=x$. Substituting: $g(x^3+1)=\sqrt[3]{(x^3+1)-1}=x$.
$g(f(x))=x$. Substituting: $g(x^3+1)=\sqrt[3]{(x^3+1)-1}=x$.
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What is $g(f(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?
What is $g(f(x))$ if $f(x)=3x$ and $g(x)=\frac{x}{3}$?
Tap to reveal answer
$g(f(x))=x$. Substituting: $g(3x)=\frac{3x}{3}=x$.
$g(f(x))=x$. Substituting: $g(3x)=\frac{3x}{3}=x$.
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What is $f(g(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?
What is $f(g(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?
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$f(g(x))=x$. Substituting: $f(2x+1)=\frac{(2x+1)-1}{2}=x$.
$f(g(x))=x$. Substituting: $f(2x+1)=\frac{(2x+1)-1}{2}=x$.
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What is $g(f(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?
What is $g(f(x))$ if $f(x)=\frac{x-1}{2}$ and $g(x)=2x+1$?
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$g(f(x))=x$. Substituting: $g(\frac{x-1}{2})=2\cdot\frac{x-1}{2}+1=x$.
$g(f(x))=x$. Substituting: $g(\frac{x-1}{2})=2\cdot\frac{x-1}{2}+1=x$.
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Identify $f(g(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
Identify $f(g(x))$ if $f(x)=e^x$ and $g(x)=\ln(x)$.
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$f(g(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.
$f(g(x))=x$ for $x>0$. Domain restriction needed for $\ln(x)$.
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What is $f(g(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $g$?
What is $f(g(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $g$?
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$f(g(x))=x$ for $x\ge 0$. Substituting: $f(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.
$f(g(x))=x$ for $x\ge 0$. Substituting: $f(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.
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What is $g(f(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $f$?
What is $g(f(x))$ if $f(x)=\sqrt{x}$ and $g(x)=x^2$ with domain $x\ge 0$ for $f$?
Tap to reveal answer
$g(f(x))=x$ for $x\ge 0$. Substituting: $g(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.
$g(f(x))=x$ for $x\ge 0$. Substituting: $g(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.
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What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?
What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?
Tap to reveal answer
$f(g(x))=x$ for $x\ge 0$. Substituting: $f(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.
$f(g(x))=x$ for $x\ge 0$. Substituting: $f(\sqrt{x})=(\sqrt{x})^2=x$ when $x\ge 0$.
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What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?
What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=\sqrt{x}$ with domain restriction $x\ge 0$ for $f$?
Tap to reveal answer
$g(f(x))=x$ for $x\ge 0$. Substituting: $g(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.
$g(f(x))=x$ for $x\ge 0$. Substituting: $g(x^2)=\sqrt{x^2}=x$ when $x\ge 0$.
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