If f(x) =log e `(1 + x)/(1 - x)` and `g(x) = (3x + x^3)/(1 + 3x^2)`, then prove that f[g(x)] = `3 cdot f(x)`.

If f(x)=log_(3)xandg(x)=x^(2) , then prove that f(g(3)) = 2.

If f(x)=log[(1+x)/(1-x)], then prove that f[(2x)/(1+x^2)]=2f(x)dot

If f(x)=2^x the f(x+3)/ f(x-1) =?

If 3f(x)-f(1/x)=log_(e)x^(4)(xgt0) , then prove that f(e^(x))=x .

If f(x) =cot ^(-1) ((3x -x^(3))/(1-3x^(2))) and g (x) =cos ^(-1) ((1- x ^(2))/(1+x^(2))) then lim _(xtoa) (f(x) -f(a))/(g (x) -g(a)) is equal to-

If f(x-2)=2x^2+3x-5 , find f(x) and prove that f(-1) = 0.

If g (x) is the inverse of f (x) and f(x)=(1)/(1+x^(3)) , then find g(x) .

If 2f(x-1)-f((1-x)/x)=x , then prove that 3f(x)=2(x+1)+1/(x+1) .

If f(x)=1+x^2 and [g(x)]=1+x^2-2x^3+x^4, g(2)=2 , then

Use the Factor Theorem to determine whether g (x) is factor of f(x) in each of the following cases: (i) f (x) = 5x ^(3) + x ^(2)-5x -1, g (x)=x +1 (ii) f (x) = x ^(3) + 3x ^(2) + 3x +1 , g(x) =x +1 (iii) f (x) =x ^(3) - 4x ^(2) +x + 6, g (x) =x -2 (iv) f (x) = 3x ^(3) - 20x + 12, g (x) = 3x -2 (v) f (x) = 4x ^(3) + 20 x ^(2) + 33 x +18, g (x) =2x +3