Show that the Signum Function f : R &rar | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

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.

 If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?

The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Determine order and degree(if defined) of differential equation y' + 5y = 0

y = x² + 2x + C : y^' - 2x - 2 = 0