The graph of a function y = f(x) is shown in the figure. One more function y = g(x) is defined as g \left(\right. x \left.\right) = \left{\right. f \left(\right. x \left.\right) - 2 , \textrm{ }\textrm{ }\textrm{ } x < 0 \\ f \left(\right. x \left.\right) , \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } x = 0 \\ \frac{- 1}{f \left(\right. x \left.\right)} \textrm{ } - \textrm{ } \frac{1}{2} , \textrm{ }\textrm{ }\textrm{ }\textrm{ } 0 \textrm{ } < \textrm{ } x < \frac{3}{2} \\ - f \left(\right. x \left.\right) , \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } x \textrm{ } \geq \textrm{ } \frac{3}{2} \textrm{ }\textrm{ }\textrm{ } Identify the correct statement(s)? [Note : [k] denotes greatest integer less than or equal to k]

Engineering

Mathematics

Introduction to Functions

Basic Definition of Continuity

Graphs Related Problems in Function

Question

The graph of a function y = f(x) is shown in the figure. One more function y = g(x) is defined as $g (x) = {\begin{matrix} f (x) - 2, x < 0 \\ f (x), x = 0 \\ \frac{- 1}{f (x)} - \frac{1}{2}, 0 < x < \frac{3}{2} \\ - f (x), x \geq \frac{3}{2} \end{matrix}$ Identify the correct statement(s)? [Note : [k] denotes greatest integer less than or equal to k]

g(x) is continuous at x = 0.

Range of g(x) is [– 4, ∞).

g(x) is discontinuous at x = ± $\frac{3}{2}$ .

$\underset{x \to 0^{-}}{Lim} [g (x)] = - 1$

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Graph of y = g(x)
From the graph it is clear that
(A) range of g(x) is [– 4, ∞)

(B) $\underset{x \to 0^{-}}{Lim} [g (x)] = - 1$
(C) g(x) is continuous at x = 0

(D) g(x) is continuous at x = $\frac{- 3}{2}$
but discontinuous at x = $\frac{3}{2}$