Question

determine where $f(x)$ is increasing. $$f(x)=|x+1|$$

   determine where $f(x)$ is increasing.

$$f(x)=|x+1|$$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 1, Problem 49

Instant Answer

Solved by Expert Linh Vu

03/03/2020

Step 1

The absolute value function $|x|$ is a V-shaped graph that is always non-negative. When we replace $x$ with $x+1$, the graph shifts to the left by 1 unit. Show more…

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determine where $f(x)$ is increasing. $$f(x)=|x+1|$$

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Key Concepts

Critical Points

Critical points are values in the domain of a function where the behavior changes, such as where the derivative is zero or undefined. For absolute value functions, the point where the expression inside the absolute value equals zero is pivotal, as it often marks the transition between intervals with different slopes, influencing where the function is increasing or decreasing.

Monotonicity

Monotonicity examines where a function is increasing or decreasing. A function is increasing on an interval if larger input values correspond to larger output values. Understanding this concept involves analyzing the slope or rate of change on each interval, which is particularly important for piecewise functions like those involving absolute values.

Absolute Value Function

An absolute value function returns the non-negative magnitude of a real number, meaning it outputs the distance of the input from zero. This concept is key in understanding how these functions reflect negative input values and create a distinct vertex where the behavior of the function changes.

Piecewise Function Analysis

Many absolute value functions can be expressed as piecewise functions, with different expressions for different portions of their domain. This piecewise representation is fundamental to analyzing function properties such as continuity, differentiability, and monotonicity, since the behavior can vary across different intervals.

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