1.

Let`f(x)=x^2`and`g(x)" "=" "2x" "+" "1`betwo real functions. Find `(f" "+" "g)``(x)`, `(f" "" "g)" "(x)`, `(fg)" "(x)`, `(f/g)(x)`.

Answer» Here, dom (f)=R and dom (g) =R.
`:."dom "(f)nn"dom "(g)=(RnnR)=R`.
(i) `(f+g):RtoR` is given by
`(f+g)(x)=f(x)+g(x)=x^(2)+(2x+1)=(x+1)^(2)`.
(ii) `(f-g):RtoR` is given by
`(f-g)(x)=f(x)-g(x)=x^(2)-(2x+1)=(x^(2)-2x-1)`.
(iii) `(fg):RtoR` is given by
`(fg)(x)=f(x).g(x)=x^(2).(2x+1)=(2x^(3)+x^(2))`
(iv) `{x:g(x)=0}={x:2x+1=0}={(-1)/(2)}`.
`:."dom "((f)/(g))=RnnR-{(-1)/(2)}=R-{(-1)/(2)}`.
The function `(f)/(g):R-{(-1)/(2)}toR` is given by
`((f)/(g))(x)=(f(x))/(g(x))=(x^(2))/(2x+1),xne(-1)/(2)`.


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