If F ( X ) = ( X ′ ′ ′ + 9 ) 2 F(x) = \left(x^{\prime \prime \prime} + 9\right)^2 F ( X ) = ( X ′′′ + 9 ) 2 , Which Statement About F ( X F(x F ( X ] Is True?A. F ( X F(x F ( X ] Is An Even Function For All Values Of M M M .B. F ( X F(x F ( X ] Is An Even Function For All Even Values Of

Introduction to Even Functions

In mathematics, an even function is a function where $f(-x) = f(x)$ for all x in the domain of the function. This means that if we replace x with -x in the function, the function remains unchanged. Even functions have a number of important properties, including symmetry about the y-axis and the fact that their graphs are mirror images of each other about the y-axis.

The Given Equation

The given equation is $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$. To determine whether this function is even, we need to substitute -x for x in the equation and see if the function remains unchanged.

Substituting -x for x

Let's substitute -x for x in the equation:

$f(-x) = \left((-x)^{\prime \prime \prime} + 9\right)^2$

Evaluating the Expression

To evaluate the expression, we need to remember that the exponentiation operation has a higher precedence than the addition operation. This means that we need to evaluate the expression inside the parentheses first.

$(-x)^{\prime \prime \prime} = -x^{\prime \prime \prime}$

Simplifying the Expression

Now we can simplify the expression:

$f(-x) = \left(-x^{\prime \prime \prime} + 9\right)^2$

Comparing the Original and Modified Equations

Now we can compare the original equation with the modified equation:

$f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$

$f(-x) = \left(-x^{\prime \prime \prime} + 9\right)^2$

Conclusion

From the above equations, we can see that $f(-x) \neq f(x)$ for all values of x. This means that the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is not an even function for all values of x.

The Correct Statement

However, we can see that if $x^{\prime \prime \prime}$ is an even function, then $f(x)$ will be an even function. This is because the square of an even function is also an even function.

The Final Answer

Therefore, the correct statement is:

B. $f(x)$ is an even function for all even values of $x^{\prime \prime \prime}$.

Additional Information

It's worth noting that the statement A is incorrect because $f(x)$ is not an even function for all values of $x^{\prime \prime \prime}$. The function $f(x)$ is only an even function if $x^{\prime \prime \prime}$ is an even function.

Conclusion

In conclusion, the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is not an even function for all values of x. However, it is an even function for all even values of $x^{\prime \prime \prime}$.

Q: What is an even function?

A: An even function is a function where $f(-x) = f(x)$ for all x in the domain of the function. This means that if we replace x with -x in the function, the function remains unchanged.

Q: What are some properties of even functions?

A: Even functions have a number of important properties, including symmetry about the y-axis and the fact that their graphs are mirror images of each other about the y-axis.

Q: Is the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ an even function for all values of x?

A: No, the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is not an even function for all values of x. However, it is an even function for all even values of $x^{\prime \prime \prime}$.

Q: Why is the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ not an even function for all values of x?

A: The function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is not an even function for all values of x because $f(-x) \neq f(x)$ for all values of x. This is due to the fact that $(-x)^{\prime \prime \prime} = -x^{\prime \prime \prime}$, which means that the function is not symmetric about the y-axis.

Q: What is the condition for the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ to be an even function?

A: The function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is an even function if $x^{\prime \prime \prime}$ is an even function.

Q: What is the significance of the exponent $x^{\prime \prime \prime}$ in the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$?

A: The exponent $x^{\prime \prime \prime}$ determines whether the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is an even function or not. If $x^{\prime \prime \prime}$ is an even function, then the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is also an even function.

Q: Can the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ be an odd function?

A: No, the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ cannot be an odd function. This is because the square of a function is always non-negative, which means that the function cannot have the symmetry properties of an odd function.

Q: What is the relationship between the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ and the function $f(-x) = \left(-x^{\prime \prime \prime} + 9\right)^2$?

A: The function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ and the function $f(-x) = \left(-x^{\prime \prime \prime} + 9\right)^2$ are related by the fact that $f(-x) \neq f(x)$ for all values of x. This means that the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is not symmetric about the y-axis.

Q: Can the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ be a periodic function?

A: Yes, the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ can be a periodic function if $x^{\prime \prime \prime}$ is a periodic function. This means that the function will repeat itself after a certain period of time.

Q: What is the significance of the constant 9 in the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$?

A: The constant 9 in the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ does not affect the symmetry properties of the function. However, it does affect the value of the function at certain points.

Q: Can the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ be a linear function?

A: No, the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ cannot be a linear function. This is because the square of a function is always non-linear.

Q: What is the relationship between the function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ and the function $f(x) = x^{\prime \prime \prime}$?

A: The function $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ and the function $f(x) = x^{\prime \prime \prime}$ are related by the fact that $f(x) = \left(x^{\prime \prime \prime} + 9\right)^2$ is a transformation of the function $f(x) = x^{\prime \prime \prime}$.