\begin{align} Let\;\; cos^{-1}\left(-\frac{1}{2}\right)=y, \;\;Then,\;\; cos y = -\frac{1}{2} = - cos\left(\frac{\pi}{3}\right)=cos\left(\pi - \frac{\pi}{3}\right) = cos\left(\frac{2\pi}{3}\right)\end{align}
We know that the range of the principal value branch of cos−1 is
\begin{align} \left[0,\pi\right] and \;\;cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\end{align}
Therefore, the principal value of
\begin{align} cos^{-1}\left(-\frac{1}{2}\right) is \frac{2\pi}{3}\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.
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by
Represent graphically a displacement of 40 km, 30° east of north.