Determining the Symmetry of the Given Function

Oct 13, 2025 by recoveryshake.com 49 views

Introduction

In mathematics, the symmetry of a function is a crucial concept that helps us understand its behavior and properties. A function can be classified as even, odd, or neither based on its symmetry properties. In this article, we will determine algebraically whether the given function $h(x)=\frac{8 x}{4 x^2-8}$ is even, odd, or neither.

What are Even and Odd Functions?

Before we dive into the problem, let's recall the definitions of even and odd functions.

Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$ . This means that the function is symmetric with respect to the y-axis.
Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ . This means that the function is symmetric with respect to the origin.

Determining the Symmetry of the Given Function

To determine whether the given function $h(x)=\frac{8 x}{4 x^2-8}$ is even, odd, or neither, we need to substitute $-x$ into the function and simplify.

Step 1: Substitute $-x$ into the function

We will substitute $-x$ into the function $h(x)=\frac{8 x}{4 x^2-8}$ and simplify.

import sympy as sp
x = sp.symbols('x')
h = (8x) / (4x**2 - 8)
h_neg_x = h.subs(x, -x)
print(h_neg_x)

Step 2: Simplify the expression

We will simplify the expression obtained in the previous step.

# Simplify the expression
h_neg_x_simplified = sp.simplify(h_neg_x)
print(h_neg_x_simplified)

Step 3: Compare the simplified expression with the original function

We will compare the simplified expression with the original function to determine whether the function is even, odd, or neither.

# Compare the simplified expression with the original function
if h_neg_x_simplified == -h:
    print("The function is odd.")
elif h_neg_x_simplified == h:
    print("The function is even.")
else:
    print("The function is neither even nor odd.")

Conclusion

In this article, we determined algebraically whether the given function $h(x)=\frac{8 x}{4 x^2-8}$ is even, odd, or neither. We substituted $-x$ into the function, simplified the expression, and compared it with the original function. Based on the comparison, we concluded that the function is neither even nor odd.

Final Answer

Introduction

In our previous article, we determined algebraically whether the given function $h(x)=\frac{8 x}{4 x^2-8}$ is even, odd, or neither. In this article, we will answer some frequently asked questions related to determining the symmetry of a function.

Q: What is the difference between an even and an odd function?

A: An even function is a function that is symmetric with respect to the y-axis, meaning that $f(-x) = f(x)$ for all $x$ in the domain of $f$ . An odd function is a function that is symmetric with respect to the origin, meaning that $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .

Q: How do I determine whether a function is even, odd, or neither?

A: To determine whether a function is even, odd, or neither, you need to substitute $-x$ into the function and simplify. If the simplified expression is equal to the original function, then the function is even. If the simplified expression is equal to the negative of the original function, then the function is odd. If the simplified expression is neither equal to the original function nor the negative of the original function, then the function is neither even nor odd.

Q: What are some examples of even and odd functions?

A: Some examples of even functions include:

$f(x) = x^2$
$f(x) = \cos x$
$f(x) = |x|$

Some examples of odd functions include:

$f(x) = x^3$
$f(x) = \sin x$
$f(x) = x$

Q: Can a function be both even and odd?

A: No, a function cannot be both even and odd. If a function is even, then it must satisfy the condition $f(-x) = f(x)$ for all $x$ in the domain of $f$ . If a function is odd, then it must satisfy the condition $f(-x) = -f(x)$ for all $x$ in the domain of $f$ . These two conditions are mutually exclusive, so a function cannot be both even and odd.

Q: Why is it important to determine whether a function is even, odd, or neither?

A: Determining whether a function is even, odd, or neither is important because it helps us understand the behavior and properties of the function. For example, even functions have symmetry with respect to the y-axis, while odd functions have symmetry with respect to the origin. This information can be useful in a variety of applications, such as physics, engineering, and computer science.

Conclusion

In this article, we answered some frequently asked questions related to determining the symmetry of a function. We discussed the difference between even and odd functions, how to determine whether a function is even, odd, or neither, and provided some examples of even and odd functions. We also addressed the question of whether a function can be both even and odd, and explained why it is important to determine the symmetry of a function.

Final Answer

The final answer is that determining the symmetry of a function is an important concept in mathematics that helps us understand the behavior and properties of a function. By understanding whether a function is even, odd, or neither, we can gain valuable insights into its behavior and properties.