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Chapter 11: Problem 10

Determine whether the function is even, odd, or neither. $$ f(x)=2|x|-1 $$

Short Answer

Expert verified
The function is even.

Step by step solution

01

Understand the Definitions

A function is even if for all values of \(x\), \(f(x) = f(-x)\). It is odd if \(f(-x) = -f(x)\). If it doesn't satisfy either condition, it is neither even nor odd.
02

Substitute \(-x\) into the Function

Substitute \(-x\) into the function \(f(x) = 2|x| - 1\) to find \(f(-x)\). This gives: \(f(-x) = 2|-x| - 1\). Since \(|x| = |-x|\), this simplifies to \(f(-x) = 2|x| - 1\).
03

Compare \(f(x)\) and \(f(-x)\)

We found that \(f(-x) = 2|x| - 1\). Notice that \(f(x) = 2|x| - 1\) is the same as \(f(-x)\). Therefore, \(f(x) = f(-x)\), which satisfies the condition for an even function.

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always non-negative. For any real number \(x\), we denote the absolute value as \(|x|\). For example, \(|3| = 3\) and \(|-3| = 3\) because both 3 and -3 are 3 units away from zero.
  • The expression \(|x| = |-x|\) holds true for every \(x\) because absolute value considers only magnitude, not direction.
  • This property is extensively used in functions to ensure a uniform output, irrespective of the sign of the input.
When dealing with functions involving absolute values, it is important to recognize that they impose symmetry on the function across the y-axis. This is because \(|x|\) reflects negative values of \(x\) into the positive side of the number line.
Function Properties
Understanding the properties of functions is crucial in determining their nature. For functions, key properties include whether they are even, odd, or neither. This characterization helps in understanding the function's graph and symmetry.
  • An even function is symmetric about the y-axis. Mathematically, a function \(f(x)\) is even if for all \(x\), \(f(x) = f(-x)\).
  • An odd function, conversely, exhibits symmetry about the origin, satisfying \(f(-x) = -f(x)\) for all \(x\).
  • If a function does not meet these conditions, it is categorized as neither even nor odd.
Knowing these properties helps predict the function's behavior without graphing. This is beneficial in simplifying equations and solving problems efficiently.
Symmetry in Functions
Symmetry in functions is a visual and mathematical concept that helps define the behavior of functions. Symmetry can be axial, point-based, or radial, but for functions, we mostly deal with axial and origin symmetry.
  • In the context of functions, symmetry about the y-axis suggests that the function evaluates to the same value for both positive and negative inputs. This is characteristic of even functions, like the one seen in the example \(f(x) = 2|x| - 1\).
  • Origin symmetry implies that the negative input of a function yields the negative output of the function's positive input, indicative of odd functions. This is not the case for our example function.
Recognizing these symmetries allows for quick decisions in calculus and algebra, such as performing integrations or analyzing the range and domain of functions. These symmetries also play a crucial role in simplifying and solving equations by reducing the complexity of calculations involved.

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Most popular questions from this chapter

Determine whether the function is even, odd, or neither. $$ f(x)=\left|x^{4}\right| $$

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