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Chapter 7: Problem 7

Graph the function \(f(x)=\sqrt{x^{2}}\). What other equation produces the same graph?

Short Answer

Expert verified
The graph of \(f(x) = \sqrt{x^2}\) is the absolute value function \(|x|\), which is identical to the piecewise function combining \(-x\) for \(x < 0\) and \(x\) for \(x \geq 0\).

Step by step solution

01

Simplify the Function Under Square Root

The given function is \(f(x) = \sqrt{x^2}\). To simplify, recall that for any real number \(x\), \(\sqrt{x^2} = |x|\). Thus, the function simplifies to \(f(x) = |x|\).
02

Understand the Absolute Value Function

The function \(f(x) = |x|\) can be understood by analyzing its behavior: for \(x \geq 0\), \(f(x) = x\) and for \(x < 0\), \(f(x) = -x\). This creates a 'V' shape graph with a vertex at the origin (0,0).
03

Determine Alternate Equation

An alternative equation that produces the same graph as \(|x|\) is \(f(x) = -x\) for \(x < 0\) and \(f(x) = x\) for \(x \geq 0\), which is the piecewise function capturing the behavior of \(|x|\).
04

Confirm the Graph

Graph both equations, \(f(x) = |x|\) and the piecewise function, to verify that they produce the same graph. Both will yield identical 'V' shaped graphs symmetric about the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Simplification
Square root simplification is an essential skill when dealing with equations involving square roots. Simplifying expressions under a square root can often make solving the equation or graphing the function much easier.

For the function given in the exercise, \(f(x) = \sqrt{x^2}\), the goal is to simplify it into a form that is easier to analyze. Let's break down the process:
  • The property \(\sqrt{x^2} = |x|\) holds for all real numbers \(x\). This is because the square root of a square value gives the absolute value of \(x\).
  • This simplification is crucial as it replaces a potentially more complex square root with the absolute value, \(f(x) = |x|\).
By simplifying \(\sqrt{x^2}\) to \(|x|\), we've formed a new function that is much simpler to understand and graph. This step highlights how simplification helps us rewrite functions into an easier format, maintaining the original function鈥檚 properties.
Piecewise Functions
Piecewise functions are an efficient way to describe a function that behaves differently based on the input values. They are particularly useful for graphing absolute value functions, such as \(f(x) = |x|\). Here鈥檚 how piecewise functions work:
  • The function \(f(x) = |x|\) can be expressed as a piecewise function. For \(x \geq 0\), \(f(x) = x\), and for \(x < 0\), \(f(x) = -x\).
  • This means that the function has two different "rules" or expressions, depending on the range of \(x\) values.
Understanding the piecewise nature of a function helps in visualizing how the function behaves under different scenarios. For absolute value functions like \(f(x) = |x|\), it results in a graph that connects the two pieces, making a 'V' shaped curve. The existence of such alternative forms, like piecewise structures, is important for recognizing the continuity and behavior of complex functions.
Symmetry in Graphs
Symmetry is a key property when analyzing graphs of functions. It simplifies the graphing process and helps understand the function's behavior. Let鈥檚 explore the symmetry in the graph of \(f(x) = |x|\).

  • The graph of \(f(x) = |x|\) is symmetric about the y-axis because for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\).
  • This means that the left and right sides of the graph are mirror images of each other. Such symmetry is a hallmark of functions that include absolute values or even components in their equations.
  • Symmetrical graphs simplify analysis since you only need to understand half the graph. The other half will be a reflection.
For this exercise, identifying symmetry confirms that both \(f(x) = |x|\) and the equivalent piecewise function give the same visual result. By recognizing these patterns, one can better interpret and graph mathematical functions.

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Most popular questions from this chapter

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