y = Ax : xy' = y (x ≠ 0)
y = Ax
Differentiating both sides with respect to x, we get:
\begin{align}y^{'}=\frac{d}{dx}(Ax)\end{align}
⇒ y ' = A
Substituting the value of y' in the given differential equation, we get:
L.H.S. = xy' = xA = Ax = y = R.H.S.
Hence, the given function is the solution of the corresponding differential equation.
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.
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible
with .