Question

Let f : W → W be defined as $f(n)={\binom{n+1,{\mathrm{if~}}~n{\mathrm{~is~even}}}{n-1,{\mathrm{~if~}}~n{\mathrm{~is~odd}}}}$ show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.

Find the solution of the differential equation $(x d y-y d x)\,y\,\mathrm{sin}{\left({\frac{y}{x}}\right)}=(y d x+x d y)x\cos\biggl({\frac{y}{x}}\biggr).$

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point $(x_{1},\;y_{1}),$ where x₁ = 2 and y₁ > 0.

Find the particular solution of the differential equation $\log\left(\frac{d y}{d x}\right)=3x+4y,$ given that y = 0, when x = 0.

If f(x) =$\frac{4x+3}{6x-4}\,,\;\;x\;\;\neq\;\;\frac{2}{3}\,,$ then Show that fof(x) = x for all $x\neq{\frac{2}{3}}\,.$ What is the inverse of f?