Understanding Odd and Even Functions

Oct 13, 2025 by miodeya.com 105 views

If $g(x)$ is an Odd Function, Which Function Must be an Even Function?

Understanding Odd and Even Functions

In mathematics, functions can be classified into two categories: odd and even functions. An odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain, while an even function satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain. Understanding the properties of odd and even functions is crucial in various mathematical and scientific applications.

Given an Odd Function $g(x)$

If $g(x)$ is an odd function, then we know that $g(-x) = -g(x)$ for all $x$ in its domain. We are asked to determine which of the given functions must be an even function.

Option A: $f(x) = g(x) + 2$

Let's analyze Option A: $f(x) = g(x) + 2$ . To determine if this function is even, we need to check if $f(-x) = f(x)$ for all $x$ in its domain. Substituting $-x$ into the function, we get:

$f(-x) = g(-x) + 2$

Since $g(x)$ is an odd function, we know that $g(-x) = -g(x)$ . Substituting this into the equation above, we get:

$f(-x) = -g(x) + 2$

Comparing this with the original function $f(x) = g(x) + 2$ , we can see that $f(-x) \neq f(x)$ for all $x$ in its domain. Therefore, Option A is not an even function.

Option B: $f(x) = g(x) + g(x)$

Let's analyze Option B: $f(x) = g(x) + g(x)$ . To determine if this function is even, we need to check if $f(-x) = f(x)$ for all $x$ in its domain. Substituting $-x$ into the function, we get:

$f(-x) = g(-x) + g(-x)$

Since $g(x)$ is an odd function, we know that $g(-x) = -g(x)$ . Substituting this into the equation above, we get:

$f(-x) = -g(x) - g(x)$

$f(-x) = -2g(x)$

Comparing this with the original function $f(x) = g(x) + g(x)$ , we can see that $f(-x) \neq f(x)$ for all $x$ in its domain. Therefore, Option B is not an even function.

Option C: $f(x) = g(x)^2$

Let's analyze Option C: $f(x) = g(x)^2$ . To determine if this function is even, we need to check if $f(-x) = f(x)$ for all $x$ in its domain. Substituting $-x$ into the function, we get:

$f(-x) = g(-x)^2$

Since $g(x)$ is an odd function, we know that $g(-x) = -g(x)$ . Substituting this into the equation above, we get:

$f(-x) = (-g(x))^2$

$f(-x) = g(x)^2$

Comparing this with the original function $f(x) = g(x)^2$ , we can see that $f(-x) = f(x)$ for all $x$ in its domain. Therefore, Option C is an even function.

Option D: $f(x) = -g(x)$

Let's analyze Option D: $f(x) = -g(x)$ . To determine if this function is even, we need to check if $f(-x) = f(x)$ for all $x$ in its domain. Substituting $-x$ into the function, we get:

$f(-x) = -g(-x)$

Since $g(x)$ is an odd function, we know that $g(-x) = -g(x)$ . Substituting this into the equation above, we get:

$f(-x) = -(-g(x))$

$f(-x) = g(x)$

Comparing this with the original function $f(x) = -g(x)$ , we can see that $f(-x) \neq f(x)$ for all $x$ in its domain. Therefore, Option D is not an even function.

Conclusion

In conclusion, the only function that must be an even function is Option C: $f(x) = g(x)^2$ . This is because when we substitute $-x$ into the function, we get $f(-x) = g(x)^2$ , which is equal to the original function $f(x) = g(x)^2$ . Therefore, Option C satisfies the condition for an even function, and it is the correct answer.

References

[1] "Odd and Even Functions" by Math Open Reference
[2] "Properties of Odd and Even Functions" by Wolfram MathWorld
[3] "Odd and Even Functions in Calculus" by Khan Academy
Q&A: Odd and Even Functions

Understanding Odd and Even Functions

In our previous article, we discussed the properties of odd and even functions. An odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain, while an even function satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain. In this article, we will answer some frequently asked questions about odd and even functions.

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

A: The main difference between an odd function and an even function is the way they behave when we substitute $-x$ into the function. An odd function satisfies the condition $f(-x) = -f(x)$ , while an even function satisfies the condition $f(-x) = f(x)$ .

Q: Can a function be both odd and even?

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

Q: How do I determine if a function is odd or even?

A: To determine if a function is odd or even, you can substitute $-x$ into the function and see if it satisfies the condition for an odd function ( $f(-x) = -f(x)$ ) or an even function ( $f(-x) = f(x)$ ).

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

A: Some examples of odd functions include:

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

Some examples of even functions include:

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

Q: Can a function be neither odd nor even?

A: Yes, a function can be neither odd nor even. For example, the function $f(x) = x^2 + 1$ is neither odd nor even, because it does not satisfy either of the conditions for an odd or even function.

Q: What are the properties of odd and even functions?

A: Odd functions have the following properties:

$f(-x) = -f(x)$ for all $x$ in its domain
$f(0) = 0$
$f(x) = -f(-x)$ for all $x$ in its domain

Even functions have the following properties:

$f(-x) = f(x)$ for all $x$ in its domain
$f(0) = f(x)$ for all $x$ in its domain
$f(x) = f(-x)$ for all $x$ in its domain

Q: How do odd and even functions behave under composition?

A: When we compose two functions, the resulting function may be odd, even, or neither. For example, if we compose two odd functions, the resulting function will be odd. If we compose two even functions, the resulting function will be even. If we compose an odd function and an even function, the resulting function will be neither odd nor even.

Conclusion

In conclusion, odd and even functions are important concepts in mathematics, and understanding their properties is crucial for solving problems in calculus, algebra, and other areas of mathematics. We hope that this article has helped to clarify the concepts of odd and even functions and has provided a useful resource for students and professionals alike.

References

[1] "Odd and Even Functions" by Math Open Reference
[2] "Properties of Odd and Even Functions" by Wolfram MathWorld
[3] "Odd and Even Functions in Calculus" by Khan Academy