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Question Number 192048 by sciencestudentW last updated on 06/May/23

$$ifthecombinedfunctionish\left(x\right)=\sqrt{3x^2+1}$$$$thenfindthetowotherfunctionsofits.$$

Answered by mehdee42 last updated on 07/May/23

$$suppose;f\left(x\right)=\sqrt{ax+b};a\ne 0,g\left(x\right)=a^{\prime }{x}^2+b^{\prime }x+c^{\prime };a^{\prime }\ne 0$$$$fog\left(x\right)=f\left(g\left(x\right)\right)\Rightarrow \sqrt{a(a^{\prime }{x}^2+b^{\prime }x+c^{\prime })+b}\equiv \sqrt{3x^2+1}$$$$\Rightarrow {aa}^{{}^{\prime }}=3\mathrm{\&}{ab}^{\prime }=0\mathrm{\&}{ac}^{\prime }+b=1$$$$fordiffrentvaluesa,b,a^{\prime },b^{\prime },c^{\prime }variuosfunctions$$$$canbeobtained.forexample:$$$$a=1,b=1,a^{\prime }=3,b^{\prime }=0\mathrm{\&}{c}^{\prime }=0$$$$\Rightarrow f\left(x\right)=\sqrt{x+1}\mathrm{\&}g\left(x\right)=3x^2$$$$or;a=3,b=-2,a^{\prime }=1,b^{\prime }=0\mathrm{\&}{c}^{\prime }=1$$$$\Rightarrow f\left(x\right)=\sqrt{3x-2}\mathrm{\&}g\left(x\right)=x^2+1$$$$andsoon$$$$$$

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