). Consider function f: R-{-1,1} →R . f(x)=(x)/(1-|x|). Then, the incorrect statement is (a) it

step 1 in this problem for x not equal to minus 1 we have fx equals to 1 minus 2 x plus 3x square and s

). Consider function f: R-{-1,1} →R . f(x)=(x)/(1-|x|)....

). Consider function $f: R-\{-1,1\} \rightarrow R . f(x)=\frac{x}{1-|x|}$.
Then, the incorrect statement is
(a) it is continuous at the origin
(b) it is not derivable at the origin
(c) the range of the function is $R$
(d) $f$ is continuous and derivable in its dor-

Key Concepts

Continuity of Functions

This concept refers to whether a function behaves without interruption at a point or within an interval. A function is continuous at a point if the limit as the input approaches that point equals the function's value there. Continuity ensures that there are no breaks or jumps in the graph of the function, which is essential for analyzing its overall behavior.

Differentiability

Differentiability concerns whether a function has a well-defined derivative at a point, meaning it has a unique tangent line there. While continuity is necessary for differentiability, a function can be continuous at a point but not differentiable if it features sharp turns or cusps. This concept is crucial when studying the smoothness and local behavior of functions.

Range of a Function

The range is the set of all possible output values produced by a function. It is important to analyze the function's range to understand the scope and limits of its outputs, particularly how the function behaves near its critical points or as the input approaches values where the function's behavior changes significantly.

Domain Restrictions

Domain restrictions denote the limitations or exclusions placed on the input values for which the function is defined. These restrictions often arise from operations that are undefined for certain inputs, such as division by zero or taking absolute value constraints, and they play a critical role in determining the valid input space for analysis.

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