If F(x) = 2x + 5 And G(x) = X^2 + 1 Be Two Real Functions, Then Describe ...

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SolveGuidesJoin / LoginUse appLogin0You visited us 0 times! Enjoying our articles? Unlock Full Access!Standard XIIMathsComposition of FunctionsQuestionIf $$f(x)=2x+5$$ and $$g(x)={x}^{2}+1$$ be two real functions, then describe given functions:(i) $$f\circ g$$(ii) $$g\circ f$$(iii) $$f\circ f$$(iv) $${f}^{2}$$Also, show that $$f\circ f\ne {f}^{2}$$.Open in AppSolutionVerified by Toppr

$$f\circ g$$ means $$g(x)$$ function is in $$f(x)$$ function. $$g\circ f$$ means $$f(x)$$ function is in $$g(x)$$ function.$$f(x)$$ and $$g(x)$$ are polynomials.$$\Rightarrow$$ $$f:R\rightarrow R$$ and $$g:R\rightarrow R.$$So, $$f\circ g:R\rightarrow R,$$ and $$g\circ f:R\rightarrow R$$$$(i)$$ $$(f\circ g)(x)=f[g(x)]$$ $$=f(x^2+1)$$......... [ Since, $$g(x)=x^2+1$$ ] $$=2(x^2+1)+5$$......... [ Since, $$f(x)=2x+5$$ ] $$=2x^2+2+5$$ $$=2x^2+7$$$$(ii)$$ $$(g\circ f)(x)=g[f(x)]$$ $$=g(2x+5)$$ ......... [ Since, $$f(x)=2x+5$$ ] $$=g(2x+5)^2+1$$ ......... [ Since, $$g(x)=x^2+1$$ ] $$=4x^2+20x+26$$$$(iii)$$ $$(f\circ f)(x)=f[f(x)]$$ $$=f(2x+5)$$ ......... [ Since, $$f(x)=2x+5$$ ] $$=2(2x+5)+5$$ $$=4x+10+5$$ $$=4x+15$$ ---- ( 1 )$$(iv)$$ $$f^2(x)=f(x)\times f(x)$$ $$=(2x+5)(2x+5)$$ $$=(2x+5)^2$$ $$=4x^2+20x+25$$ ----- ( 2 )From ( 1 ) and ( 2 ),$$\Rightarrow$$ $$(f\circ f)(x)\ne f^2(x)$$$$\therefore$$ $$f\circ f\ne f^2$$

Was this answer helpful?2Similar QuestionsQ1If $$f(x)=2x+5$$ and $$g(x)={x}^{2}+1$$ be two real functions, then describe given functions:(i) $$f\circ g$$(ii) $$g\circ f$$(iii) $$f\circ f$$(iv) $${f}^{2}$$Also, show that $$f\circ f\ne {f}^{2}$$.View SolutionQ2If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv) f2 Also, show that fof ≠ f2View SolutionQ3

Let f, g be two real functions defined by f(x)=√x+1 and g(x)=√9−x2. Then, describe each of the following functions:

(i) f+g (ii) g−f (iii) fg (iv) fg (v) gf (vi) 2f−√5g (vii) f2+7f (viii) 58

View SolutionQ4If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv) f2 Also, show that fof ≠ f2View SolutionQ5Let $$f$$ be a real function given by $$f(x)=\sqrt{x-2}$$. Find the following.(i) $$f\circ f$$ (ii) $$f\circ f\circ f$$ (iii) $$(f\circ f\circ f)(38)$$ (iv) $${f}^{2}$$Also, show that $$f\circ f\ne {f}^{2}$$.View Solution

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