Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible
with .
f: R+ → [ - 5, ∞) is given as f(x) = 9x2 + 6x - 5.
Let y be an arbitrary element of [ - 5, ∞).
Let y = 9x2 + 6x - 5.
∴f is onto, thereby range f = [ - 5, ∞).
Let us define g: [ - 5, ∞) → R+ as
We now have:
∴ gof = IR+ and
Hence, f is invertible and the inverse of f is given by
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.
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Determine order and degree(if defined) of differential equation ym + 2yn + y' =0
Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by , where R+ is the set of all non-negative real numbers.
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
The order of the differential equation
\begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}
is (A) 2 (B) 1 (C) 0 (D) not defined