Which Statement Best Describes How To Determine Whether $f(x)=x^4-x^3$ Is An Even Function?A. Determine Whether $(-x)^4 - (-x)^3$ Is Equivalent To $x^4 - X^3$.B. Determine Whether $\left(-x^4\right) -

Understanding 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 symmetry about the y-axis and are often represented by functions with even powers of x.

Determining Even Functions: A Step-by-Step Guide

To determine whether a function is even, we need to check if $f(-x) = f(x)$. Let's consider the given function $f(x) = x^4 - x^3$. To determine whether this function is even, we need to evaluate $f(-x)$ and compare it with $f(x)$.

Option A: Evaluating $(-x)^4 - (-x)^3$

One way to determine whether $f(x) = x^4 - x^3$ is an even function is to evaluate $(-x)^4 - (-x)^3$ and compare it with $x^4 - x^3$. Let's start by expanding $(-x)^4$ and $(-x)^3$.

$(-x)^4 = (-x)(-x)(-x)(-x) = x^4$

$(-x)^3 = (-x)(-x)(-x)(-x) = -x^3$

Now, let's substitute these values into the expression $(-x)^4 - (-x)^3$.

$(-x)^4 - (-x)^3 = x^4 - (-x^3)$

$(-x)^4 - (-x)^3 = x^4 + x^3$

As we can see, $(-x)^4 - (-x)^3$ is not equivalent to $x^4 - x^3$. Therefore, option A is not the correct way to determine whether $f(x) = x^4 - x^3$ is an even function.

Option B: Evaluating $\left(-x^4\right) - \left(-x^3\right)$

Another way to determine whether $f(x) = x^4 - x^3$ is an even function is to evaluate $\left(-x^4\right) - \left(-x^3\right)$ and compare it with $x^4 - x^3$. Let's start by expanding $\left(-x^4\right)$ and $\left(-x^3\right)$.

$\left(-x^4\right) = -x^4$

$\left(-x^3\right) = -x^3$

Now, let's substitute these values into the expression $\left(-x^4\right) - \left(-x^3\right)$.

$\left(-x^4\right) - \left(-x^3\right) = -x^4 + x^3$

As we can see, $\left(-x^4\right) - \left(-x^3\right)$ is not equivalent to $x^4 - x^3$. Therefore, option B is not the correct way to determine whether $f(x) = x^4 - x^3$ is an even function.

Option C: Evaluating $f(-x)$

To determine whether $f(x) = x^4 - x^3$ is an even function, we need to evaluate $f(-x)$ and compare it with $f(x)$. Let's start by substituting $-x$ into the function.

$f(-x) = (-x)^4 - (-x)^3$

$f(-x) = x^4 - (-x^3)$

$f(-x) = x^4 + x^3$

As we can see, $f(-x)$ is not equivalent to $f(x)$. Therefore, the function $f(x) = x^4 - x^3$ is not an even function.

Conclusion

In conclusion, to determine whether a function is even, we need to evaluate $f(-x)$ and compare it with $f(x)$. The correct way to determine whether $f(x) = x^4 - x^3$ is an even function is to evaluate $f(-x)$ and compare it with $f(x)$. The function $f(x) = x^4 - x^3$ is not an even function.

Key Takeaways

• An even function is a function where $f(-x) = f(x)$ for all x in the domain of the function.
• To determine whether a function is even, we need to evaluate $f(-x)$ and compare it with $f(x)$.
• The function $f(x) = x^4 - x^3$ is not an even function.

Final Answer

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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: How do I determine whether a function is even?

A: To determine whether a function is even, you need to evaluate $f(-x)$ and compare it with $f(x)$. If $f(-x) = f(x)$, then the function is even.

Q: What is the correct way to determine whether $f(x) = x^4 - x^3$ is an even function?

A: The correct way to determine whether $f(x) = x^4 - x^3$ is an even function is to evaluate $f(-x)$ and compare it with $f(x)$. This is option C in the previous article.

Q: Why is $f(x) = x^4 - x^3$ not an even function?

A: $f(x) = x^4 - x^3$ is not an even function because $f(-x)$ is not equivalent to $f(x)$. When we substitute $-x$ into the function, we get $f(-x) = x^4 + x^3$, which is not equal to $f(x) = x^4 - x^3$.

Q: What are some common characteristics of even functions?

A: Even functions have several common characteristics, including:

• Symmetry about the y-axis
• Even powers of x
• $f(-x) = f(x)$ for all x in the domain of the function

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 $f(-x) = f(x)$ for all x in the domain of the function. If a function is odd, then $f(-x) = -f(x)$ for all x in the domain of the function. These two properties are mutually exclusive.

Q: How do I determine whether a function is odd?

A: To determine whether a function is odd, you need to evaluate $f(-x)$ and compare it with $-f(x)$. If $f(-x) = -f(x)$, then the function is odd.

Q: What is the relationship between even and odd functions?

A: Even and odd functions are related in that they can be combined to form a new function. Specifically, if $f(x)$ is an even function and $g(x)$ is an odd function, then $f(x) + g(x)$ is an odd function, and $f(x) - g(x)$ is an even function.

Q: Can you give an example of an even function?

A: Yes, a simple example of an even function is $f(x) = x^2$. When we substitute $-x$ into the function, we get $f(-x) = (-x)^2 = x^2$, which is equal to $f(x)$. Therefore, $f(x) = x^2$ is an even function.

Q: Can you give an example of an odd function?

A: Yes, a simple example of an odd function is $f(x) = x^3$. When we substitute $-x$ into the function, we get $f(-x) = (-x)^3 = -x^3$, which is equal to $-f(x)$. Therefore, $f(x) = x^3$ is an odd function.