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Chapter 2: Problem 25

What type of symmetry does an even function have?

Short Answer

Expert verified
An even function has y-axis symmetry.

Step by step solution

01

- Define an Even Function

An even function is defined as a function that satisfies the property \( f(-x) = f(x) \) for all x in its domain.
02

- Understand Symmetry

Symmetry in mathematics refers to a situation where one half of an object or function is a mirror image of the other half.
03

- Identify the Type of Symmetry for Even Functions

Since an even function satisfies the equation \( f(-x) = f(x) \), it means that the function is mirrored around the y-axis. This is known as y-axis symmetry.

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Key Concepts

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

even function
Understanding what an even function is quite simple once you get the hang of its core principle. An even function satisfies the rule: \( f(-x) = f(x) \) for all x in its domain.
This means that when you input the negative of a number, the output is the same as when you input the positive of that number.
For instance, if \( f(2) = 5 \), an even function guarantees \( f(-2) = 5 \) as well.
This property helps in identifying and working with even functions in various mathematical problems.
Examples of even functions include:
  • \( f(x) = x^2 \)
  • \( f(x) = \text{cos}(x) \)
  • \( f(x) = x^4 \)
Recognizing even functions can greatly simplify calculations and interpretations.
symmetry in mathematics
Symmetry in mathematics is a fundamental concept in all areas of math, from geometry to algebra to advanced calculus.
It essentially describes when one part of a figure or function is a mirror image of another part.
There are different types of symmetries, with y-axis symmetry (related to even functions) being just one example.
Other types include:
  • **Rotational Symmetry**: Where an object looks the same after a certain amount of rotation.
  • **Reflective Symmetry**: Where one half is the mirror reflection of the other half.
  • **Translational Symmetry**: When a pattern can be moved or 'translated' along a certain direction and still look the same.
Understanding symmetry helps make sense of complex shapes and makes solving equations easier, since we can exploit repeated patterns.
y-axis symmetry
Y-axis symmetry specifically means that a graph is mirrored around the y-axis.
In simpler terms, the left side of the graph is a mirror image of the right side.
For any function to have y-axis symmetry, it must satisfy the condition of being an even function, which is \( f(-x) = f(x) \).
Let's consider the function \( f(x) = x^2 \):
  • When x = 2, \( f(2) = 2^2 = 4 \).
  • When x = -2, \( f(-2) = (-2)^2 = 4 \).
Here, you see that \( f(2) = f(-2) \), meaning the function is symmetrical about the y-axis.
This concept helps in graphing functions since knowing one side automatically informs you about the other side.

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Most popular questions from this chapter

The water level in a retention pond started at \(5 \mathrm{ft}(60 \mathrm{in} .)\) and decreased at a rate of 2 in./day during a 14 -day drought. A tropical depression moved through at the beginning of the 15 th day and produced rain at an average rate of 2.5 in./day for 5 days. a. Write a piecewise-defined function to model the water level \(L(x)\) (in inches) as a function of the number of days \(x\) since the beginning of the drought. b. Graph the function.

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A cell phone plan charges \(\$ 49.95\) per month plus \(\$ 14.02\) in taxes, plus \(\$ 0.40\) per minute for calls beyond the 600 -min monthly limit. Write a piecewise-defined function to model the monthly cost \(C(x)\) (in \$) as a function of the number of minutes used \(x\) for the month.
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