For some a, b, c $\in\mathbb{N}$, let $f(x) = ax - 3$ and $\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$. If ${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}$, then $(fog)(ac) + (gof)(b)$ is equal to ____________.
Answer (integer)
2039
Solution
$f(x)=a x-3$
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$g(x)=x^{b}+c$
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$(fog)^{-1}=\left(\frac{x-7}{2}\right)^{\frac{1}{3}}$
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$(fog)^{-1}(x)=\left(\frac{x+3-c a}{a}\right)^{\frac{1}{b}}=\left(\frac{x-7}{2}\right)^{\frac{1}{3}}$
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$\Rightarrow a=2, b=3, c=5$
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$fog(a c)+gof(b)$
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$\because f(x)=2 x-3$
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$g(x)=x^{3}+5$
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$fog(10)+g o f(3)$
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$=2007+32$
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$=2039$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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