Which Of The Following Is An Odd Function?A. F ( X ) = X 3 + 5 X 2 + X F(x) = X^3 + 5x^2 + X F ( X ) = X 3 + 5 X 2 + X B. F ( X ) = X F(x) = \sqrt{x} F ( X ) = X ​ C. F ( X ) = X 2 + X F(x) = X^2 + X F ( X ) = X 2 + X D. F ( X ) = − X F(x) = -x F ( X ) = − X

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Identifying Odd Functions: A Mathematical Analysis

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). Functions can be classified into different types based on their properties, such as even or odd functions. In this article, we will discuss the concept of odd functions and identify which of the given functions is an odd function.

An odd function is a function that satisfies the condition:

f(-x) = -f(x)

for all x in the domain of the function. In other words, if we replace x with -x in the function, the resulting function should be equal to the negative of the original function. This property is a key characteristic of odd functions.

Odd functions have several important properties:

  • Symmetry: Odd functions are symmetric with respect to the origin. This means that if we reflect the graph of the function about the y-axis, the resulting graph will be the same as the original graph.
  • Periodicity: Odd functions are periodic with a period of 2π. This means that the graph of the function repeats itself every 2π units.
  • Zero at the origin: Odd functions have a zero at the origin, i.e., f(0) = 0.

Some common examples of odd functions include:

  • f(x) = x^3
  • f(x) = x^5
  • f(x) = sin(x)
  • f(x) = tan(x)

Now, let's analyze the given functions to determine which one is an odd function.

A. f(x) = x^3 + 5x^2 + x

To determine if this function is odd, we need to check if f(-x) = -f(x). Let's substitute -x into the function:

f(-x) = (-x)^3 + 5(-x)^2 + (-x) = -x^3 + 5x^2 - x

Comparing this with -f(x), we get:

-f(x) = -(x^3 + 5x^2 + x) = -x^3 - 5x^2 - x

Since f(-x) ≠ -f(x), this function is not an odd function.

B. f(x) = √x

To determine if this function is odd, we need to check if f(-x) = -f(x). Let's substitute -x into the function:

f(-x) = √(-x)

Since the square root of a negative number is not a real number, this function is not defined for all real numbers. Therefore, it is not an odd function.

C. f(x) = x^2 + x

To determine if this function is odd, we need to check if f(-x) = -f(x). Let's substitute -x into the function:

f(-x) = (-x)^2 + (-x) = x^2 - x

Comparing this with -f(x), we get:

-f(x) = -(x^2 + x) = -x^2 - x

Since f(-x) ≠ -f(x), this function is not an odd function.

D. f(x) = -x

To determine if this function is odd, we need to check if f(-x) = -f(x). Let's substitute -x into the function:

f(-x) = -(-x) = x

Comparing this with -f(x), we get:

-f(x) = -(-x) = x

Since f(-x) = -f(x), this function is an odd function.

In conclusion, the function f(x) = -x is the only odd function among the given options. This function satisfies the condition f(-x) = -f(x) for all x in the domain of the function. The other functions do not satisfy this condition and are therefore not odd functions.

  • [1] "Odd Function." MathWorld, Wolfram Research, 2023.
  • [2] "Even and Odd Functions." Khan Academy, 2023.
  • [3] "Properties of Odd Functions." MIT OpenCourseWare, 2023.
    Odd Functions: A Q&A Guide =============================

In our previous article, we discussed the concept of odd functions and identified which of the given functions is an odd function. In this article, we will answer some frequently asked questions about odd functions to help you better understand this mathematical concept.

A: An even function is a function that satisfies the condition f(-x) = f(x) for all x in the domain of the function. On the other hand, an odd function is a function that satisfies the condition f(-x) = -f(x) for all x in the domain of the function.

A: Some common examples of odd functions include:

  • f(x) = x^3
  • f(x) = x^5
  • f(x) = sin(x)
  • f(x) = tan(x)
  • f(x) = -x

A: Some common examples of even functions include:

  • f(x) = x^2
  • f(x) = cos(x)
  • f(x) = e^x
  • f(x) = |x|

A: The graph of an odd function is symmetric with respect to the origin. This means that if we reflect the graph of the function about the y-axis, the resulting graph will be the same as the original graph.

A: The period of an odd function is 2π. This means that the graph of the function repeats itself every 2π units.

A: An odd function has a zero at the origin, i.e., f(0) = 0.

A: Yes, an odd function can be a polynomial function. For example, f(x) = x^3 + 5x^2 + x is an odd function.

A: Yes, an odd function can be a trigonometric function. For example, f(x) = sin(x) is an odd function.

A: Yes, an odd function can be a rational function. For example, f(x) = -x is an odd function.

A: Yes, an odd function can be a composite function. For example, f(x) = sin(x^3) is an odd function.

In conclusion, odd functions are an important concept in mathematics, and understanding their properties and examples can help you better analyze and solve mathematical problems. We hope this Q&A guide has helped you understand odd functions better.

  • [1] "Odd Function." MathWorld, Wolfram Research, 2023.
  • [2] "Even and Odd Functions." Khan Academy, 2023.
  • [3] "Properties of Odd Functions." MIT OpenCourseWare, 2023.