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.
If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f . g)oh = (foh) . (goh)
Represent graphically a displacement of 40 km, 30° east of north.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.
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}
The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?