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Question

Find fog and gof if
(i) fx=ex, gx=loge x
(ii) fx=x2, gx=cos x
(iii) fx=|x|, g (x)=sin x
(iv) fx=x+1, gx=ex
(v) fx=sin-1 x, gx=x2
(vi) fx=x+1, gx=sin x
(vii) fx=x+1, gx=2x+3
(viii) fx=c, c R, gx=sin x2
(ix) fx=x2+2, gx=1-11-x

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Solution

i f x=ex, gx=loge xf:R0,; g:0,RComputing fog:Clearly, the range of g is a subset of the domain of f.fog : 0,Rfog x=f g x=f loge x=loge ex=xComputing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g ex=loge ex=x

ii f x=x2, gx=cos xf:R[0, ) ; g:R-1, 1Computing fog:Clearly, the range of g is not a subset of the domain of f.Domain fog=x: xdomain of g and gxdomain of fDomain fog=x: xR and cos x R}Domain of fog=Rfog: RRfog x=f g x=f cos x=cos2xComputing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g x2=cos x2


iii f x=x, gx=sin xf:R0, ; g:R-1, 1Computing fog:Clearly, the range of g is a subset of the domain of f.fog : RRfog x=f g x=f sin x=sin xComputing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g x=sin x

iv f x=x+1, gx=exf:RR; g:R[1, )Computing fog:Clearly, range of g is a subset of domain of f.fog : RRfog x=f g x=f ex=ex+1Computing gof:Clearly, range of f is a subset of domain of g.fog : RRgof x=g f x=g x+1=ex+1

v f x=sin-1x, gx=x2f:-1,1-π2,π2 ; g:R[0, )Computing fog:Clearly, the range of g is not a subset of the domain of f.Domain fog=x: xdomain of g and gxdomain of fDomain fog=x: xR and x2-1,1Domain fog=x: xR and x-1,1Domain of fog=-1,1fog: -1,1Rfog x=f g x=f x2=sin-1 x2Computing gof:Clearly, the range of f is a subset of the domain of g.fog : -1,1Rgof x=g f x=g sin-1x=sin-1 x2

vi fx=x+1, gx=sin xf:RR ; g:R-1, 1Computing fog:Clearly, the range of g is a subset of the domain of f.fog: RRfog x=f g x=f sin x=sin x+1Computing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g x+1=sin x+1

vii f x=x+1, gx=2x+3f:RR ; g:RRComputing fog:Clearly, the range of g is a subset of the domain of f.fog: RRfog x=f g x=f 2x+3=2x+3+1=2x+4Computing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g x+1=2 x+1+3=2x+5

viii f x=c, gx=sin x2f:Rc ; g:R0, 1Computing fog:Clearly, the range of g is a subset of the domain of f.fog: RRfog x=f g x=f sin x2=cComputing gof:Clearly, the range of f is a subset of the domain of g.fog : RRgof x=g f x=g c=sin c2

ix fx=x2+2f:R[2,) gx=1-11-xFor domain of g: 1-x0 x1Domain of g=R-1gx=1-11-x=1-x-11-x=-x1-xFor range of g:y=-x1-xy-xy=-xy=xy-xy=xy-1x=yy-1Range of g =R-1So, g: R-1R-1Computing fog:Clearly, the range of g is a subset of the domain of f.fog: R-1Rfog x=f g x=f -xx-1=-xx-12+2=x2+2x2+2-4x1-x2=3x2-4x+21-x2Computing gof:Clearly, the range of f is a subset of the domain of g.gof : RRgof x=g f x=g x2+2=1-11-x2+2=1-1-x2+1=x2+2x2+1

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