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Chapter 11: Problem 95

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

Short Answer

Expert verified
The function is neither even nor odd.

Step by step solution

01

- Understand Even and Odd Functions

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

- Substitute \( x \) with \( -x \) in the Function

Given the function \( f(x) = -x^{5} \), substitute \( x \) with \( -x \) to get \( f(-x) \). Thus \( f(-x) = -(-x)^{5} \).
03

- Simplify \( f(-x) \)

Simplify the expression \( f(-x) = -(-x)^{5} \). Since \( (-x)^{5} = -x^{5} \), then \( f(-x) = -(-x^{5}) = x^{5} \).
04

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

Now compare \( f(-x) = x^{5} \) with \( f(x) = -x^{5} \). Since \( f(-x) \) is not equal to \( f(x) \) and \( f(-x) \) is not equal to \( -f(x) \), 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 has a specific property: its graph is symmetric with respect to the y-axis. The formal definition is that a function is even if for every x in its domain, the equation \( f(x) = f(-x) \) holds true. For example, the function \( f(x) = x^2 \) is even because when you replace x with -x, you still get \( f(x) = (-x)^2 = x^2 \). Even functions have graphs that look the same on both sides of the y-axis.
odd functions
Odd functions also have a unique symmetry. They are symmetric with respect to the origin. This means if you rotate the graph 180 degrees around the origin, it looks the same. The formal definition states that a function is odd if for every x in its domain, the equation \( f(-x) = -f(x) \) is true. For instance, the function \( f(x) = x^3 \) is odd because \( f(-x) = (-x)^3 = -x^3 = -f(x) \). Notice how the sign changes, indicating the function's nature as odd.
polynomial symmetry
Polynomial symmetry helps determine whether a polynomial function is even, odd, or neither. You can use substitution to explore the symmetry. By replacing \( x \) with \( -x \) in the polynomial function, you can check if the new function equals the original or its negative. For example, given \( f(x) = -x^5 \), substituting \( -x \) gives \( f(-x) = -(-x)^5 = -(-x^5) = x^5 \). Since this new function (\( x^5 \)) neither matches \( f(x) \) nor \( -f(x) \), the function is neither even nor odd. Checking symmetry this way simplifies identifying the function's properties.
function properties
Function properties provide insights into how functions behave. The two main characteristics relevant here are whether a function is even or odd. However, it's also important to recognize functions that don't fit these categories. Some functions, like \( f(x) = -x^5 \), are neither even nor odd and lack specific symmetry. Recognizing these properties involves comparing \( f(x) \), \( f(-x) \), and \( -f(x) \) to identify any symmetry. Knowing the properties allows for a better understanding of how to analyze and graph functions effectively. It simplifies many problems in calculus and algebra.

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Most popular questions from this chapter

Approximate all real zeros of each function to the nearest hundredth. \(f(x)=-2.47 x^{3}-6.58 x^{2}-3.3 x+0.1\)

Use a graphing calculator to find (or approximate) the real zeros of each function \(f(x)\). Express decimal approximations to the nearest hundredth. \(f(x)=4 x^{4}+8 x^{3}-4 x^{2}+4 x+1\)

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)?

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

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