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Chapter 11: Problem 98

Determine whether each polynomial function is even, odd, or neither. \(f(x)=4 x^{5}-x^{4}\)

Short Answer

Expert verified
The function is neither even nor odd.

Step by step solution

01

- Understanding Definitions

A function is even if \( f(-x) = f(x) \) for all x in the domain. A function is odd if \( f(-x) = -f(x) \) for all x in the domain. If neither of these conditions are met, the function is neither even nor odd.
02

- Substitute -x into the Function

Substitute \( -x \) for \( x \) into the function \( f(x) \). This gives us \( f(-x) = 4(-x)^{5} - (-x)^{4} \).
03

- Simplify the Expression

Simplify \( f(-x) \) to see if we can identify if it's equal to \( f(x) \) or \(-f(x) \). This results in \( f(-x) = 4(-x)^{5} - (-x)^{4} = 4(-x^5) - x^4 = -4x^5 - x^4 \).
04

- Compare f(-x) with f(x)

Compare the simplified \( f(-x) = -4x^5 - x^4 \) to the original \( f(x) = 4x^5 - x^4 \). Since \( f(-x) \) is not equal to \( f(x) \) or \ -f(x) \, the function is neither even nor odd.

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Key Concepts

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

Even Functions
An even function is one where the graph is symmetrical about the y-axis. This means that for any x in the function's domain, if you substitute -x, the value of the function remains the same. Mathematically, this is represented as: f(-x) = f(x).

For example, the function f(x) = x^2 is even because:
  • f(-x) = (-x)^2 = x^2 = f(x), demonstrating symmetry about the y-axis.
When working with polynomial functions, you can quickly identify even functions by observing the powers of x. If all the powers in the polynomial are even, and all the coefficients are real numbers, the function is even.

Recognizing even functions helps simplify many calculus problems, especially those involving integration over symmetric intervals.
Odd Functions
An odd function displays symmetry about the origin. This means that substituting -x into the function changes the sign of the value of the function. The mathematical condition for an odd function is: f(-x) = -f(x).

Consider the function f(x) = x^3. It's an odd function because:
  • f(-x) = (-x)^3 = -x^3 = -f(x), which shows central symmetry about the origin.
In polynomial functions, if all the powers of x are odd, and all coefficients are real numbers, the function is classified as odd.

Identifying odd functions is crucial in various calculus problems, like simplifying integrals over specific intervals where symmetry about the origin can be advantageous.
Function Symmetry
Function symmetry is a key concept in understanding even and odd functions. Symmetry helps to determine whether a function is even or odd.
  • Y-axis symmetry: A function is symmetrical about the y-axis if flipping across the y-axis results in the same graph. This is the property of even functions.
  • Origin symmetry: A function is symmetrical about the origin if rotating the graph 180 degrees results in the same graph. This is the property of odd functions.
Knowing these symmetries helps in sketching graphs and solving integrals by making use of symmetrical properties to reduce complexity and computation. Understanding symmetry also helps in categorizing solutions efficiently.
Substitution Method
The substitution method is widely used in evaluating if polynomial functions are even or odd. This involves substituting -x into the function and analyzing the result.

For the given exercise, substitute -x for x in the function f(x) = 4x^5 - x^4:
f(-x) = 4(-x)^5 - (-x)^4 = 4(-x^5) - x^4 = -4x^5 - x^4.

After substitution and simplification, compare the new expression with the original.
  • If f(-x) = f(x), the function is even.
  • If f(-x) = -f(x), the function is odd.
  • If it matches neither of these, then the function is neither even nor odd.
Utilizing the substitution method simplifies the determination of function symmetry and classification.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This is crucial when assessing function characteristics.

In our exercise, the substitution leads to evaluating:
  • 4(-x)^5 - (-x)^4 = -4x^5 - x^4
Each term was simplified by following the rules of exponents and then comparing it to the original function.
Simplification helps in accurately determining the function's properties:
  • For even and odd functions, simplification ensures correct identification.
  • It reduces the complexity of polynomials.
  • Simplification aids in better understanding and visualizing the function's behavior.
Being adept at simplifying expressions is fundamental in mathematics, serving various applications in calculus and algebra alike.

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Most popular questions from this chapter

Concept Check The graphs of four polynomial functions are shown in \(A-D .\) They represent the graphs of functions defined by these four equations, but not necessarily in the order listed. $$ \begin{array}{ll} y=x^{3}-3 x^{2}-6 x+8 & y=x^{4}+7 x^{3}-5 x^{2}-75 x \\ y=-x^{3}+9 x^{2}-27 x+17 & y=-x^{5}+36 x^{3}-22 x^{2}-147 x-90 \end{array} $$ Apply the concepts of this section to answer each question. Which of the graphs cannot be that of a cubic polynomial function?

The table shows the total (cumulative) number of ebola cases reported in Sierra Leone during a serious West African ebola outbreak in \(2014-2015 .\) The total number of cases is reported \(x\) months after the start of the outbreak in May \(2014 .\) $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Months after } \\ \text { May 2014 } \end{array} & \text { Total Ebola Cases } \\ \hline 0 & 16 \\ 2 & 533 \\ 4 & 2021 \\ 6 & 7109 \\ 8 & 10,518 \\ 10 & 11,841 \\ 12 & 12,706 \\ 14 & 13,290 \\ 16 & 13,823 \\ 18 & 14,122 \\ \hline \end{array} $$ (a) Use the regression feature of a calculator to determine the quadratic function that best fits the data. Let \(x\) represent the number of months after May \(2014,\) and let \(y\) represent the total number of ebola cases. Give coefficients to the nearest hundredth. (b) Repeat part (a) for a cubic function (degree 3). Give coefficients to the nearest hundredth. (c) Repeat part (a) for a quartic function (degree 4). Give coefficients to the nearest hundredth. (d) Compare the correlation coefficient \(R^{2}\) for the three functions in parts (a)-(c) to determine which function best fits the data. Give its value to the nearest ten-thousandth.

Use a graphing calculator to find (or approximate) the real zeros of each function \(f(x)\). Express decimal approximations to the nearest hundredth. \(f(x)=2.45 x^{4}-3.22 x^{3}+0.47 x^{2}-6.54 x+3\)

Determine whether each polynomial function is even, odd, or neither. \(f(x)=2 x^{3}\)

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier sections. (a) What is the common factor in the numerator and the denominator? (b) For what value of \(x\) will there be a point of discontinuity (a hole)?
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