Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
Let f: X → Y be an invertible function.
Then, there exists a function g: Y → X such that gof = Ix and fog = Iy
Here, f-1 = g.
Now, gof = Ix and fog = Iy
⇒ f-1of = Ix and fof-1= Iy
Hence, f-1: Y → X is invertible and f is the inverse of f-1
i.e., (f-1)-1 = f.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
Represent graphically a displacement of 40 km, 30° east of north.
If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
The degree of the differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
is (A) 3 (B) 2 (C) 1 (D) not defined
In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Determine order and degree(if defined) of differential equation yn + (y')2 + 2y =0
Determine order and degree(if defined) of differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
A balloon, which always remains spherical, has a variable diameter
\begin{align} \frac{3}{2}(2x+1)\end{align}
Find the rate of change of its volume with respect to x.