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Chapter 11: Problem 91

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

Short Answer

Expert verified
The function f(x) = 2x^3 is odd.

Step by step solution

01

- Recall definitions

A function is even if for every x in the domain, f(x) = f(-x). A function is odd if f(-x) = -f(x). If neither condition is satisfied, the function is neither even nor odd.
02

- Substitute -x into the function

Substitute -x into the given function f(x) = 2x^3 thought the expression f(-x): f(-x) = 2(-x)^3.
03

- Simplify the expression

Simplify the expression for f(-x): f(-x) = 2(-x)^3 = 2(-x^3) = -2x^3.
04

- Compare with f(x)

Compare the simplified expression of f(-x) with f(x): f(-x) = -2x^3 and f(x) = 2x^3. Notice that f(-x) = -f(x).
05

- Determine function type

Since f(-x) = -f(x), f(x) = 2x^3 is an odd function.

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Key Concepts

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

even function
In mathematics, an even function has specific properties that make it unique. The core property of an even function is that it satisfies the equation \(f(x) = f(-x)\) for every value of \(x\) in its domain. This means that if you plug in \(-x\) into the function, the output will be the same as if you had plugged in \(x\). A common example is the polynomial function \(f(x) = x^2\). When you substitute \(-x\), you get: \[f(-x) = (-x)^2 = x^2 = f(x)\]This symmetry means the graph of an even function is mirrored about the y-axis. Recognizing an even function is crucial because it helps to understand its behavior and apply it in various mathematical and real-world problems.
odd function
Odd functions have a characteristic property where substituting \(-x\) into the function results in the negation of the original function. Mathematically, an odd function satisfies \(f(-x) = -f(x)\). To see this, consider the polynomial function \(f(x) = x^3\). When substituting \(-x\), you get: \[f(-x) = (-x)^3 = -x^3 = -f(x)\]This property indicates that the graph of an odd function is symmetric about the origin. In simpler terms, if you rotate the graph 180 degrees around the origin, it will map onto itself. Recognizing odd functions helps to predict their graphical representations and understand their physical interpretations in various contexts.
polynomial properties
Polynomials are mathematical expressions involving sums of powers of variables, typically written as \(a_nx^n + a_{n-1}x^{n-1} + \text{...} + a_1x + a_0\). Polynomials have several important properties that make them applicable in different areas of mathematics and science. Key properties include:
  • Degree: The highest power of the variable in a polynomial (e.g., degree of \(x^3 + 2x^2 + 3x + 4\) is 3).
  • Coefficients: The constants multiplying the variable terms (e.g., in \(2x^3\), the coefficient is 2).
  • Symmetry: Polynomials can be classified as even, odd, or neither based on their symmetry properties. As we discussed, \(x^2\) is even, while \(x^3\) is odd.
Polynomials also follow specific rules for addition, subtraction, and multiplication, making them easy to manipulate. Understanding these properties allows you to solve polynomial equations, graph their functions, and apply them in real-world problems such as calculating trajectories or optimizing functions.

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Most popular questions from this chapter

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) $$ f(x)=x^{3}-4 x^{2}+2 x+1 ; \quad k=-1 $$

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 $$ f(x)=2 x^{2}+5 x-12 $$ Evaluate \(f\left(\frac{3}{2}\right)\).

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.

Graph each rational function. $$f(x)=\frac{(x-5)(x-2)}{x^{2}+9}$$

Use synthetic division to determine whether the given number is a zero of the polynomial function. $$ 3 ; \quad f(x)=2 x^{3}-6 x^{2}-9 x+27 $$
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