Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f(g(x)) = 8{x^2} - 2x$ and $g(f(x)) = 4{x^2} + 6x + 1$, then the value of $f(2) + g(2)$ is _________.
Answer (integer)
18
Solution
$f(g(x))=8 x^{2}-2 x$
$g(f(x))=4 x^{2}+6 x+1$
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let $f(x)=c x^{2}+d x+e$
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$g(x)=a x+b$
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$f(g(x))=c(a x+b)^{2}+d(a x+b)+e \equiv 8 x^{2}-2 x$
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$g(f(x))=a\left(c x^{2}+d x+e\right)+b \equiv 4 x^{2}+6 x+1$
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$\therefore \quad a c=4 \quad a d=6 \quad a e+b=1$
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$a^{2} c=8 \quad 2 a b c+a d=-2 \quad c b^{2}+b d+e=0$
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By solving
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$a=2 \quad b=-1$
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$c=2 \quad d=3 \quad e=1$
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$\therefore \quad f(x)=2 x^{2}+3 x+1$
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$g(x)=2 x-1$
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$f(2)+g(2)=2(2)^{2}+3(2)+1+2(2)-1$
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$=18$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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