\begin{align} \int \left(4e^{3x} + 1\right).dx \end{align}
\begin{align} =4\int e^{3x}.dx + \int 1.dx \end{align}
\begin{align} =4\left(\frac {e^{3x}}{3}\right) + x + C \end{align}
\begin{align} =\frac {4}{3} e^{3x} + x + C \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.
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.
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)