Question

If the function f : R → R be defined by f(x) = 2x – 3 and g : R → R by g(x) = x³ + 5, then find fog and show that fog is invertible. Also, find (fog)⁻¹, hence find (fog)⁻¹(9).

Solve the differential equation given as : $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{d x}{d y}\;=1,x\neq0$

Let f, g : R → R be two functions defined as f (x) = |x| + x and g (x) = |x| – x for all x ∈ R.Then find fog and gof.

Using properties of determinants, prove that : $\begin{vmatrix}1 & 1+p & 1+p+q \\ 3 & 4+3p & 2+4p+3p \\ 4 & 7+4p & 2+7p+4q\end{vmatrix}$ = 1

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+1)}}\,.$