R = {(a, b): a ≤ b3}
It is observed that
\begin{align} \left(\frac{1}{2},\frac{1}{2}\right) ∉ R , as \frac{1}{2}>\left(\frac{1}{2}\right)^3 = \frac{1}{8}\end{align}
∴ R is not reflexive.
Now,
(1, 2) ∈ R (as 1 < 23 = 8)
But,
(2, 1) ∉ R (as 23 > 1)
∴ R is not symmetric.
We have
\begin{align} \left(3,\frac{3}{2}\right),\left(\frac{3}{2},\frac{6}{5}\right) ∉ R , as 3>\left(\frac{3}{2}\right)^3 and \frac{3}{2}<\left(\frac{6}{5}\right)^3 \end{align}
But
\begin{align} \left(3,\frac{6}{5}\right) ∉ R , as 3>\left(\frac{6}{5}\right)^3 \end{align}
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
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.
y = cosx + C : y' + sinx = 0
Find gof and fog, if
(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x1/3 .
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?