The function f(x)=(|x-1|)/(x^2) (A) increases in (-∞, 0) ∪(1,2) (B) increases in (0,1) ∪(2, ∞) (

step 1 So we can write fx equal x minus 1 upon x squared if x is greater than equal to 1 or 1 minus x u

The function f(x)=(|x-1|)/(x^2) (A) increases in (-∞, 0)...

The function $f(x)=\frac{|x-1|}{x^{2}}$
(A) increases in $(-\infty, 0) \cup(1,2)$
(B) increases in $(0,1) \cup(2, \infty)$
(C) decreases in $(0,1) \cup(2, \infty)$
(D) decreases in $(-\infty, \infty) \cup(1,2)$

Key Concepts

Differentiation and Derivative Tests

Differentiation is a key tool in analyzing the behavior of functions. Derivative tests, in particular, help determine intervals where a function is increasing or decreasing by examining the sign of its derivative.

Domain of a Function

The domain of a function consists of all possible input values for which the function is defined. Special attention is needed when a function involves division, as values that would make the denominator zero must be excluded from the domain.

Monotonicity

Monotonicity refers to the property of a function being either entirely non-increasing or non-decreasing over an interval. Determining the monotonicity of a function often involves analyzing its derivative to see where it is positive (indicating increasing behavior) or negative (indicating decreasing behavior).

Absolute Value Function

An absolute value function returns the non-negative magnitude of a number or expression, effectively splitting the function into separate cases based on whether the argument is positive or negative. This piecewise definition critically affects the behavior and properties of the function, especially when combined with other operations.

Piecewise Function

Because the absolute value leads to different expressions depending on the sign of its input, functions involving absolute values are often considered piecewise. Identifying and analyzing these pieces separately is crucial for understanding the overall behavior of the function.

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