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Chapter 11: Problem 93

Determine whether each polynomial function is even, odd, or neither. \(f(x)=0.2 x^{4}\)

Short Answer

Expert verified
The function \[f(x) = 0.2 x^4\] is even.

Step by step solution

01

Understand the Definitions

A function is even if \[f(-x) = f(x)\]and odd if \[f(-x) = -f(x)\].If neither condition is met, the function is neither even nor odd.
02

Substitute \[-x\] into \[f(x)\]

Given \[f(x) = 0.2 x^4\], substitute \[-x\] into the function: \[f(-x) = 0.2 (-x)^4\].
03

Simplify the Expression

Simplify \[f(-x) = 0.2 (-x)^4\] using the property that \[(-x)^4 = x^4\]:n\[f(-x) = 0.2 x^4\].
04

Compare \[f(-x)\] with \[f(x)\]

Notice that \[f(-x) = 0.2 x^4\] is indeed equal to \[f(x)\]. Therefore, \[f(x)\] is an even function.

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Key Concepts

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

Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It typically looks like this: \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\). The highest power of the variable is called the degree of the polynomial. For example, in the polynomial \(0.2 x^4\), the degree is 4 because the highest power of \(x\) is 4.
  • Polynomials with even degrees can be even functions or neither.
  • Polynomials with odd degrees can be odd functions or neither.
Degree and type of polynomial often indicate the behavior of the function. Understanding these properties helps to classify functions quickly and accurately.
Function Properties
Functions have certain properties that help us to describe their behavior. One important property is symmetry. A function can be classified as even, odd, or neither based on this symmetry.
  • An even function satisfies \(f(-x) = f(x)\) for all \(x\). Graphically, it is symmetric about the y-axis.
  • An odd function satisfies \(f(-x) = -f(x)\) for all \(x\). Graphically, it is symmetric about the origin.
If neither condition is met, the function is neither even nor odd. Recognizing these properties is essential to understanding and predicting the behavior of functions in various mathematical contexts.
Mathematical Definitions
Mathematical definitions provide the foundation for understanding various concepts in mathematics.
  • The definition of an even function is that \(f(-x) = f(x)\).
  • The definition of an odd function is that \(f(-x) = -f(x)\).
In our example, the polynomial function \(f(x) = 0.2 x^4\) is even because \(f(-x) = 0.2 (-x)^4 = 0.2 x^4 = f(x)\). Simplifying and verifying these definitions help determine the function's symmetry and categorize it correctly.

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Most popular questions from this chapter

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting $$ f(x)=2 x^{2}+5 x-12 $$ Solve \(f(x)=0\)

A storage tank for butane gas is to be built in the shape of a right circular cylinder of altitude \(12 \mathrm{ft}\), with a half sphere attached to each end. If \(x\) represents the radius of each half sphere, what radius should be used to cause the volume of the tank to be \(144 \pi \mathrm{ft}^{3} ?\)

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.

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=x^{3}+4 x^{2}-8 x-8, \quad[-3.8,-3]\)

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. Find the equation of the horizontal asymptote.
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