1.

Let f be the exponential function and g be the logarithumic function. Then, find (i) (f+g)(1) (ii) (fg)(1) (iii) (4f)(1) (iv) (3g)(1)

Answer» Let `f:RtoR:f(x)=e^(x)andg:R^(+)toR:g(x)=log_(e)x`.
Then, dom `(f)nn"dom "(g)=RnnR^(+)=R^(+)`.
(i) `(f+g):R^(+)toR` is given by
`(f+g)(x)=f(x)+g(x)=(e^(x)+log_(e)x)`.
`:.(f+g)(1)=(e^(1)+log_(e)1)=(e+0)=e`.
(ii) `(fg):R^(+)toR` is given by
`(fg)(x)=f(x).g(x)=e^(x)(log_(e)x)`.
`:.(fg)(1)=e^(1)(log_(1)1)=(exx0)=0`.
(iii) `(4f):R^(+)toR` is given by
`(4f)(x)=4xxf(x)=4e^(x)`.
`:.(4f)(1)=(4xxe^(1))=4e^(x)`.
(iv) `(3g):R^(+)toR` is given by
`(3g)(x)=3xxg(x)=3xx(log_(e)x)`.
`:.(3g)(1)=3xxg(1)=3xx(log_(e)1)=(3xx0)=0`.


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