Sketch the graph of f by hand and use your sketch to find...

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1/x $, $ x \geqslant 1 $

Key Concepts

Graph Sketching

Graph sketching involves creating a visual representation of a function by plotting key points, identifying trends, and examining features like intercepts, continuity, and behavior at the boundaries. This process is critical to understanding the overall behavior of a function and aids in identifying important features such as local and absolute extrema.

Domain Restrictions

Domain restrictions define the set of input values over which the function is considered. When a function is limited to a specific interval, this constraint significantly influences the potential locations of extrema and the overall behavior of the function's graph.

Reciprocal Function

A reciprocal function is characterized by the basic form f(x) = 1/x, where the output value is the multiplicative inverse of the input. This type of function typically exhibits a decreasing behavior with rapid changes for small inputs and gradual changes as the input becomes large, and often possesses asymptotic behavior.

Asymptotic Behavior

Asymptotic behavior refers to the tendency of a function to approach a particular line or value as the input grows large or nears a specific point. Identifying horizontal or vertical asymptotes helps predict the long-term behavior of functions, which is vital when analyzing functions defined by reciprocals or rational expressions.

Absolute and Local Extrema

Absolute and local extrema are the maximum and minimum values a function attains either over its entire domain or within specific intervals. These are identified by analyzing the endpoints of the domain, critical points where the derivative is zero or undefined, and the overall shape of the function's graph.

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