CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

If f(x)=loge(1x) and g(x)=[x, then determine each of the following functions :

(i) f+g
(ii) fg
(iii) fg
(iv) gf
Also, find (f+g)(1),(fg)(0),(fg)(12),(gf)(12)


Solution

We have
f(x)=loge(1x) and g(x)=[x]
f(x)=loge(1x) is defined, if 1x>0
1>xx<1xϵ(,1)
g(x) = [x] is defined for all xϵR
Domain (g) = R
Domain(f)RDomain(g)=(,1)R=(,1)

(i) f+g:(,1)R defined by
(f+g)(x)=f(x)+g(x)=loge(1x)+[x]

(ii) fg:(,1)R defined by (fg)(x)=f(x)×g(x)
=loge(1x)×[x]=[x]loge(1x)

(iii) g(x) = [x]
[x] = 0
xϵ(0,1)
So, domain (fg)=domain(f)domain(g){x:g(x)=0}=(,0)
fg:(,0) defined by (fg)(x)=loge(1x)[x]

(iv) We have,
f(x)=loge(1x)
1f(x)=1loge(1x) is defined, and loge(1x)0
1x>0 and 1x0
x<1 and x0
xϵ(,0)(0,1)
Domain (gf)=(,0)(0,1)
gf:(,0)(0,1)R defined by
(gf)(x)=[x]loge(1x)
Now,
(f+g)(1)=f(1)+g(1)=loge(1(1))+[1]=loge21(f+g)(1)=loge21

(v) fg(0)=loge(10)×[0]

(vi) (fg)(12)= does not exist

(vii) (gf)(12)=(12)loge(112)=0


Mathematics
RD Sharma
Standard XI

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
Same exercise questions
View More


similar_icon
People also searched for
View More



footer-image