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Chapter 2: Problem 6

Determine whether \(f\) is even, odd, or neither even nor odd. $$f(x)=7 x^{5}-4 x^{3}$$

Short Answer

Expert verified
The function \(f(x) = 7x^5 - 4x^3\) is odd.

Step by step solution

01

Define Even Function

A function is considered even if for every \(x\) in the domain of \(f\), \(f(-x) = f(x)\). This means the graph of the function is symmetrical with respect to the y-axis.
02

Define Odd Function

A function is considered odd if for every \(x\) in the domain of \(f\), \(f(-x) = -f(x)\). This means the graph of the function is symmetrical with respect to the origin.
03

Compute \(f(-x)\)

Substitute \(-x\) into the function \(f(x)\): \[ f(-x) = 7(-x)^5 - 4(-x)^3 = 7(-x^5) - 4(-x^3) = -7x^5 + 4x^3 \]
04

Compare \(f(x)\) and \(f(-x)\) for Even Symmetry

Since \(f(-x) = -7x^5 + 4x^3\) and \(f(x) = 7x^5 - 4x^3\), we have \(f(-x) eq f(x)\). Therefore, \(f(x)\) is not an even function.
05

Check for Odd Symmetry

To confirm if \(f(x)\) is odd, check if \(f(-x) = -f(x)\). Calculate \(-f(x)\): \[ -f(x) = -(7x^5 - 4x^3) = -7x^5 + 4x^3 \]Since this is equal to \(f(-x)\), \(f(x)\) satisfies the condition to be an odd function.

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Key Concepts

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

Symmetry in Functions
When exploring function symmetries, there are two main types:
  • Even Functions: These functions show symmetry about the y-axis. If you fold the graph along the y-axis, both sides would align perfectly.
  • Odd Functions: These have symmetry about the origin. This means if you rotate the graph 180 degrees, it would look the same.
Understanding symmetry is crucial for graphing functions and analyzing their behavior. Not all functions exhibit symmetry, but identifying even or odd functions can simplify calculations and predictions about how a function behaves across its domain.
Testing Function Parity
Testing whether a function is even, odd, or neither involves a set of straightforward steps:
  • Even Parity Test: For a function to be even, substituting \(x\) with \(-x\) must not change the function's expression: \(f(-x) = f(x)\). This suggests a symmetry along the y-axis.
  • Odd Parity Test: For a function to be odd, substituting \(x\) with \(-x\) should transform the function into its negative: \(f(-x) = -f(x)\). This indicates symmetry through the origin.
If a function fails both tests, it is neither even nor odd. Testing parity simplifies understanding function properties, and with polynomial functions, this involves checking the behavior of each term when substituting with \(-x\).
Polynomial Functions
Polynomial functions consist of terms of the form \(a_n x^n\), where \(a_n\) is a constant and \(n\) is a non-negative integer. In the context of testing for evenness or oddness:
  • Even-degree terms contribute to even functions. This means terms like \(x^2, x^4\) show no change under \(-x\).
  • Odd-degree terms contribute to odd functions. Terms like \(x^3, x^5\) switch signs when \(-x\) is substituted.
For example, in the function \(f(x) = 7x^5 - 4x^3\), both terms have odd degrees, leading to the function being odd. By recognizing the degree of terms, one can quickly determine potential symmetry, helping in both graphing and algebraic manipulation.

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