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Chapter 2: Problem 5

Determine whether \(f\) is even, odd, or neither even nor odd. $$f(x)=3 x^{4}-6 x^{2}-5$$

Short Answer

Expert verified
The function \(f(x) = 3x^4 - 6x^2 - 5\) is even.

Step by step solution

01

Understand the Definitions

A function \( f(x) \) is **even** if for every \( x \), \( f(-x) = f(x) \). A function is **odd** if for every \( x \), \( f(-x) = -f(x) \). If neither condition is met, the function is neither even nor odd.
02

Substitute \( -x \) into the Function

To determine if the function is even or odd, substitute \( -x \) into \( f(x) \) and simplify. For the given function, we have:\[ f(-x) = 3(-x)^4 - 6(-x)^2 - 5 \]This simplifies to:\[ f(-x) = 3x^4 - 6x^2 - 5 \]
03

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

Now we check if \( f(-x) = f(x) \) or \( f(-x) = -f(x) \).For \( f(x) = 3x^4 - 6x^2 - 5 \) and \( f(-x) = 3x^4 - 6x^2 - 5 \),\( f(-x) = f(x) \), proving that the function is even.

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Key Concepts

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

Even and Odd Functions
In mathematics, understanding the nature of a function involves categorizing it as even, odd, or neither. These categories offer insight into a function's behavior and symmetries. A function, denoted as \( f(x) \), is termed as **even** if substituting \( -x \) for \( x \) yields the original function: \( f(-x) = f(x) \). Conversely, a function is **odd** when \( f(-x) = -f(x) \).
  • **Even Functions**: Display symmetry about the y-axis.
  • **Odd Functions**: Exhibit rotational symmetry around the origin.
  • **Neither**: When neither condition is satisfied, functions are labeled as neither even nor odd and lack symmetric properties.
Recognizing these patterns necessitates substituting \( -x \) into the function and simplifying to assess the equality criteria. This method helps classify the function accurately.
Symmetry in Functions
Symmetry in functions is a fundamental concept that helps understand the graphical representation of functions. For functions:
  • **Y-axis Symmetry**: Even functions are symmetric about the y-axis. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\).
  • **Origin Symmetry**: Odd functions are symmetric around the origin. This implies that if \((x, y)\) lies on the graph, then \((-x, -y)\) also lies on the graph.
This symmetry helps in easily predicting and sketching the graph of the function. Knowing the symmetry saves time and reduces computation as it allows for only a section of the function to be calculated before mirroring it across the axis(if even) or around the origin(if odd). Discovering these symmetries becomes essential when dealing with complex polynomial equations or when simplifying algebraic expressions.
Polynomial Functions
Polynomial functions are a type of function defined by an expression consisting of variables and coefficients. These variables are raised to whole number exponents, and the function takes the general form: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]Each term's degree impacts the polynomial's structure and graph:
  • **Polynomial Degree**: Refers to the highest exponent of the variable in the function. It dictates the graph's shape and end behavior.
  • **Even Degree Polynomials**: More likely to exhibit y-axis symmetry, thus qualifying them as even functions, depending on their terms.
  • **Odd Degree Polynomials**: Can demonstrate origin symmetry, potentially classifying them as odd functions.
Understanding these aspects of polynomial functions enables predicting whether a polynomial function is even, odd, or neither. Tools like substitution and knowing the degree of each term within the polynomial can prompt quicker insights into their symmetrical properties.

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Most popular questions from this chapter

Quantity discount A company sells running shoes to dealers at a rate of \(\$ 40\) per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600 ), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company?

Sketch the graph of f. $$\text { 51 } f(x)=\left\\{\begin{array}{ll} x+2 & \text { if } x \leq-1 \\ x^{3} & \text { if }|x|<1 \\ -x+3 & \text { if } x \geq 1 \end{array}\right.$$

(a) Use the quadratic formula to find the zeros of \(f .\) (b) Find the maximum or minimum value of \(f(x)\) (c) Sketch the graph of \(f\) $$f(x)=-4 x^{2}+4 x-1$$

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=0.5 x^{3}-4 x-5 ; \quad g(x)=0.5 x^{3}-4 x-1$$

Graph \(y=x^{3}-x^{1 / 3}\) and \(f\) on the same \(c o-\) ordinate plane, and estimate the points of intersection. $$f(x)=x^{2}-x-\frac{1}{4}$$
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