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  • Functions f, g : R → R are defined, respectively, by f(x) = x2 + 3x + 1, g(x) = 2x – 3, find (i) fog (ii) gof (iii) fof (iv) gog

Functions f, g : R \rightarrow R are defined, respectively, by f(x) = x\textsuperscript{2} + 3x + 1, g(x) = 2x - 3, find

(i) fog (ii) gof (iii) fof (iv) gog

Answers (1)

Here,

f(x) = x\textsuperscript{2} + 3x + 1, g (x) = 2x - 3\\


\\(i)fog = f(g(x))\\ = f(2x - 3)\\ = (2x - 3)\textsuperscript{2} + 3(2x - 3) + 1\\ = 4x\textsuperscript{2}\ + 9 - 12x + 6x - 9 + 1\\ = 4x\textsuperscript{2}\ - 6x + 1\\ (ii) gof = g(f(x))\\ = g(x\textsuperscript{2}\ + 3x + 1)\\ = 2(x\textsuperscript{2}\ + 3x + 1) - 3\\ = 2x\textsuperscript{2}\ + 6x - 1\\ (iii) fof = f(f(x))\\ = f(x\textsuperscript{2}\ + 3x + 1)\\ = (x\textsuperscript{2}\ + 3x + 1)2 + 3(x\textsuperscript{2} + 3x + 1) + 1\\ = x4 + 9x\textsuperscript{2}\ + 1 + 6x\textsuperscript{3} + 6x + 2x\textsuperscript{2}\ + 3x\textsuperscript{2}\ + 9x + 3 + 1\\ = x4 + 6x\textsuperscript{3} + 14x\textsuperscript{2}\ + 15x + 5\\ (iv) gog = g(g(x))\\ = g(2x - 3)\\ = 2(2x - 3) - 3\\ = 4x - 6 - 3\\ = 4x - 9\\

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