The function f(x)=|x+2|+|x-1| is (A) increasing in (1, ∞) (B) increasing in [1, ∞) (C) decreasin

step 1 in this section function f x is given so we can write f x equal minus 2 x minus 1 when x is less

The function f(x)=|x+2|+|x-1| is (A) increasing in (1, ∞)...

Key Concepts

Critical Points in Absolute Value Analysis

Critical points are the values at which the expressions inside the absolute value functions become zero. These points divide the domain into intervals where the overall function’s behavior can change, such as switching from increasing to decreasing. Identifying and analyzing these points is essential for understanding the overall behavior of the function.

Decreasing Function

A decreasing function is one where the function’s output values become smaller as the input increases over a specific interval. Similar to increasing functions, determining if a function is decreasing involves analyzing the sign of its derivative or the slopes within the relevant intervals of its piecewise definition.

Absolute Value Function

The absolute value function returns the nonnegative magnitude of a real number. It is typically defined piecewise to handle both positive and negative inputs, which is essential for understanding functions that involve absolute values. This concept is crucial when you need to break the function into intervals where each absolute value term can be expressed as either a positive or negative linear expression.

Piecewise-defined Function

A piecewise-defined function is one that has different expressions for different intervals of its domain. When dealing with absolute value functions, rewriting the function in a piecewise manner helps analyze its behavior, such as calculating derivatives or understanding monotonicity (whether the function is increasing or decreasing) over specific intervals.

Increasing Function

An increasing function is one where the function's output values become larger as the input increases over a certain interval. Determining if a function is increasing typically involves examining its derivative or the slopes of its linear pieces in its piecewise representation.

Please give Ace some feedback