{"@context":"https://schema.org/","@type":"Quiz","name":"

The rational function $f$ is defined by an equation in the form $f(x)=\\dfrac{a}{x+b}$, where $a$ and $b$ are constants. The partial graph of $y=f(x)$ is shown. If $g(x)=f(x+4)$, which equation could define function $g$?

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","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-239"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

The rational function $f$ is defined by an equation in the form $f(x)=\\dfrac{a}{x+b}$, where $a$ and $b$ are constants. The partial graph of $y=f(x)$ is shown. If $g(x)=f(x+4)$, which equation could define function $g$?

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","comment":{"@type":"Comment","text":"

Choice C is correct. It's given that $f(x)=\\frac{a}{x+b}$ and that the graph shown is a partial graph of $y=f(x)$. Substituting $y$ for $f(x)$ in the equation $f(x)=\\frac{a}{x+b}$ yields $y=\\frac{a}{x+b}$. The graph passes through the point $(-7,-2)$. Substituting $-7$ for $x$ and $-2$ for $y$ in the equation $y=\\frac{a}{x+b}$ yields $-2=\\frac{a}{-7+b}$. Multiplying each side of this equation by $-7+b$ yields $-2(-7+b)=a$, or $14-2b=a$. The graph also passes through the point $(-5,-6)$. Substituting $-5$ for $x$ and $-6$ for $y$ in the equation $y=\\frac{a}{x+b}$ yields $-6=\\frac{a}{-5+b}$. Multiplying each side of this equation by $-5+b$ yields $-6(-5+b)=a$, or $30-6b=a$. Substituting $14-2b$ for $a$ in this equation yields $30-6b=14-2b$. Adding $6b$ to each side of this equation yields $30=14+4b$. Subtracting $14$ from each side of this equation yields $16=4b$. Dividing each side of this equation by $4$ yields $4=b$. Substituting $4$ for $b$ in the equation $14-2b=a$ yields $14-2(4)=a$, or $6=a$. Substituting $6$ for $a$ and $4$ for $b$ in the equation $f(x)=\\frac{a}{x+b}$ yields $f(x)=\\frac{6}{x+4}$. It's given that $g(x)=f(x+4)$. Substituting $x+4$ for $x$ in the equation $f(x)=\\frac{6}{x+4}$ yields $f(x+4)=\\frac{6}{x+4+4}$, which is equivalent to $f(x+4)=\\frac{6}{x+8}$. It follows that $g(x)=\\frac{6}{x+8}$.

Choice A is incorrect. This could define function $g$ if $g(x)=f(x-4)$.

Choice B is incorrect. This could define function $g$ if $g(x)=f(x)$.

Choice D is incorrect. This could define function $g$ if $g(x)=f(x)\\cdot(x+4)$.

"},"encodingFormat":"text/markdown","text":"

The rational function $f$ is defined by an equation in the form $f(x)=\\dfrac{a}{x+b}$, where $a$ and $b$ are constants. The partial graph of $y=f(x)$ is shown. If $g(x)=f(x+4)$, which equation could define function $g$?

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","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$g(x)=\\dfrac{6}{x}$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$g(x)=\\dfrac{6}{x+4}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$g(x)=\\dfrac{6(x+4)}{x+4}$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$g(x)=\\dfrac{6}{x+8}$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-239"}

The rational function $f$ is defined by an equation in the form $f(x)=\dfrac{a}{x+b}$, where $a$ and $b$ are constants. The partial graph of $y=f(x)$ is shown. If $g(x)=f(x+4)$, which equation could define function $g$?