The anti derivative of e2x is a function of x whose derivative is e2x.
It is known that,
\begin{align} \frac {d}{dx} (e^{2x}) = 2e^{2x} \end{align}
⇒ \begin{align} e^{2x} =\frac {1}{2} \frac {d}{dx}(e^{2x}) \end{align}
∴ \begin{align} e^{2x} = \frac {d}{dx}\left(\frac {1}{2}e^{2x}\right) \end{align}
Therefore, the anti derivative of e2x
\begin{align} e^{2x} \;is \frac {1}{2}e^{2x} \end{align}
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.
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.
y = Ax : xy' = y (x ≠ 0)
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by