Diberikan fungsi f ( x ) = a x − b dan g ( x ) = c x + b dengan , b ,dan adalah bilangan-bilangan real positif. Syarat agar f ( g ( x )) > g ( f ( x )) adalah …

Pembahasan

Untuk mengetahui syarat agar berlaku f ( g ( x )) > g ( f ( x )) , akan coba jabarkan bentuk dari f ( g ( x )) > g ( f ( x )) sebagai berikut. &g(f(x))\\a\left(g(x)\right)-b&>&c\left(f(x)\right)+b\\a\left(cx+b\right)-b&>&c\left(ax-b\right)+b\\{\color[rgb]{1.0, 0.0, 0.0}a}{\color[rgb]{1.0, 0.0, 0.0}c}{\color[rgb]{1.0, 0.0, 0.0}x}+ab-b&>&{\color[rgb]{1.0, 0.0, 0.0}a}{\color[rgb]{1.0, 0.0, 0.0}c}{\color[rgb]{1.0, 0.0, 0.0}x}-bc+b\\ab-b&>&-bc+b\\ab-b{\color[rgb]{0.0, 0.0, 1.0}+}{\color[rgb]{0.0, 0.0, 1.0}b}{\color[rgb]{1.0, 0.0, 0.0}+}{\color[rgb]{1.0, 0.0, 0.0}b}{\color[rgb]{1.0, 0.0, 0.0}c}&>&-bc{\color[rgb]{1.0, 0.0, 0.0}+}{\color[rgb]{1.0, 0.0, 0.0}b}{\color[rgb]{1.0, 0.0, 0.0}c}+b{\color[rgb]{0.0, 0.0, 1.0}+}{\color[rgb]{0.0, 0.0, 1.0}b}\\ab+bc&>&2b\\b\left(a+c\right)&>&2b\\a+c&>&2\\&&\end{array}" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo»(«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mi»g«/mi»«mo»(«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mfenced»«mrow»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«mo»-«/mo»«mi»b«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mi»c«/mi»«mfenced»«mrow»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mfenced»«mrow»«mi»c«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mfenced»«mo»-«/mo»«mi»b«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mi»c«/mi»«mfenced»«mrow»«mi»a«/mi»«mi»x«/mi»«mo»-«/mo»«mi»b«/mi»«/mrow»«/mfenced»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi mathcolor=¨#FF0000¨»a«/mi»«mi mathcolor=¨#FF0000¨»c«/mi»«mi mathcolor=¨#FF0000¨»x«/mi»«mo»+«/mo»«mi»a«/mi»«mi»b«/mi»«mo»-«/mo»«mi»b«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mi mathcolor=¨#FF0000¨»a«/mi»«mi mathcolor=¨#FF0000¨»c«/mi»«mi mathcolor=¨#FF0000¨»x«/mi»«mo»-«/mo»«mi»b«/mi»«mi»c«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»b«/mi»«mo»-«/mo»«mi»b«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mo»-«/mo»«mi»b«/mi»«mi»c«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»b«/mi»«mo»-«/mo»«mi»b«/mi»«mo mathcolor=¨#0000FF¨»+«/mo»«mi mathcolor=¨#0000FF¨»b«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«mi mathcolor=¨#FF0000¨»c«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mo»-«/mo»«mi»b«/mi»«mi»c«/mi»«mo mathcolor=¨#FF0000¨»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«mi mathcolor=¨#FF0000¨»c«/mi»«mo»+«/mo»«mi»b«/mi»«mo mathcolor=¨#0000FF¨»+«/mo»«mi mathcolor=¨#0000FF¨»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»b«/mi»«mo»+«/mo»«mi»b«/mi»«mi»c«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»b«/mi»«mfenced»«mrow»«mi»a«/mi»«mo»+«/mo»«mi»c«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mo»+«/mo»«mi»c«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«/mtable»«/math»" role="math" 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style="max-width: none;"> Diperoleh, syarat agar f ( g ( x )) > g ( f ( x )) berlakuadalah a + c > 2 . Jadi, jawaban yang benar adalah C.