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Chapter 11: Problem 94

Determine whether each polynomial function is even, odd, or neither. \(f(x)=-x^{6}\)

Short Answer

Expert verified
The function \( f(x) = -x^6 \) is even.

Step by step solution

01

Understand the Definitions

A function is even if for all x in the domain of the function, the equation \[ f(-x) = f(x) \] holds true. Similarly, a function is odd if for all x in the domain of the function, the equation \[ f(-x) = -f(x) \] holds true. If neither condition holds, the function is neither even nor odd.
02

Substitute \( -x \) into the Function

Given the function \( f(x) = -x^6 \), substitute \( -x \) for x: \[ f(-x) = -(-x)^6 \].
03

Simplify the Expression

Simplify \[ f(-x) = -(-x)^6 \]. Since the exponent is even, \[ (-x)^6 = x^6 \], so the expression becomes: \[ f(-x) = -x^6 \].
04

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

Observe that \[ f(-x) = -x^6 \] and \[ f(x) = -x^6 \]. Since \( f(-x) = f(x) \), the function is even.

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions composed of variables (x) raised to whole number exponents, combined using addition, subtraction, and multiplication. Each term in a polynomial function follows the form \text {Coefficient} \times \text{Variable}^{\text {Exponent}}. Some common examples include:
  • Linear: \( f(x) = ax + b \)
  • Quadratic: \( g(x) = ax^2 + bx + c \)
  • Cubic: \( h(x) = ax^3 + bx^2 + cx + d \)
The overall behavior of polynomials, such as their shape and symmetry, depends on the degree (the highest exponent) and coefficients. In the case of the exercise, the polynomial function is \( f(x) = -x^6 \). The highest exponent is 6, making it a degree-6 polynomial.
Function Symmetry
Function symmetry helps determine whether a function is even, odd, or neither. Here's a quick guide:
  • Even functions have mirror symmetry about the y-axis. For these functions, \( f(-x) = f(x) \).
  • Odd functions have rotational symmetry about the origin. For these functions, \( f(-x) = -f(x) \).
  • If a function doesn't satisfy either, it is neither even nor odd.
Substituting negative x into \( f(x) = -x^6 \), we get \( f(-x) = -(-x)^6 \). Since the sixth power of a negative number is positive, we find that \( (-x)^6 = x^6 \), thus \( f(-x) = -x^6 \). Comparing this to \( f(x) = -x^6 \), it becomes clear that \( f(-x) = f(x) \), confirming the function's even symmetry.
Algebraic Functions
Algebraic functions are functions defined using polynomials and their roots. Polynomial functions are a subset of algebraic functions. Understanding whether these functions are even or odd can lend insights into their behavior and graphing characteristics. For algebraic functions:
  • Even algebraic functions are symmetric about the y-axis, simplifying our analysis and graphing tasks.
  • Odd algebraic functions are symmetric about the origin, affecting our understanding of their zeroes and behavior across positive and negative values.
In the case of our function \( f(x) = -x^6 \), recognizing it as an even polynomial helps predict its behavior without needing to graph it extensively. Such understanding can simplify how we approach algebraic computations and solutions in various problems.

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Most popular questions from this chapter

Graph each rational function. $$f(x)=\frac{-4 x}{3 x-1}$$

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) $$ f(x)=-x^{3}+8 x^{2}+63 ; \quad k=4 $$

For each of the following, (a) show that the polynomial function has a zero between the two given integers and (b) approximate all real zeros to the nearest thousandth. \(f(x)=x^{4}+x^{3}-6 x^{2}-20 x-16 ;\) between -2 and -1

Fill in the blanks with the correct responses: By the definition of an even function, if \((a, b)\) lies on the graph of an even function, then so does \((-a, b)\). Therefore, the graph of an even function is symmetric with respect to the ____. If \((a, b)\) lies on the graph of an odd function, then by definition, so does \((-a,-b)\). Therefore, the graph of an odd function is symmetric with respect to the ____.

Use synthetic division to divide. $$ \frac{-3 x^{5}+2 x^{4}-5 x^{3}-6 x^{2}-1}{x+2} $$
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