If the function f : R → R be given by f(x) = x² + 2 and g : R → R be given by g(x) = find fog and gof and hence find fog(2) and gof(– 3).
If the function f : R → R is given by f(x) = and g : R → R is given by g(x) = 2x – 3, then find
(i) fog (ii) gof
Is f⁻¹ = g?
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) = Show that f is bijective. Also, find
(i) x, if = 4
(ii)
Show that the function f : R {x ∈ R – 1 < x < 1} defined by f(x) = x ∈ R is one-one and onto function. Hence find f⁻¹(x).
Let f : W → W be defined as show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.
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).
Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1.
Find the angle of intersection of the curves y² = 4ax and x²= 4by.
In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.
Obtain the differential equation of all the circles of radius r.
Find the particular solution of the differential equation given that y = 0 when x = 1.
Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ =
Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cos θ is at a constant distance from the origin.