Which Statement Best Describes How To Determine Whether F ( X ) = X 4 − X 3 F(x)=x^4-x^3 F ( X ) = X 4 − X 3 Is An Even Function?A. Determine Whether ( − X ) 4 − ( − X ) 3 (-x)^4-(-x)^3 ( − X ) 4 − ( − X ) 3 Is Equivalent To X 4 − X 3 X^4-x^3 X 4 − X 3 .B. Determine Whether ( − X ) 4 − ( − X ) 3 (-x)^4-(-x)^3 ( − X ) 4 − ( − X ) 3 Is Equivalent To

=====================================================

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 output remains the same. Even functions have symmetry about the y-axis, and they are often represented by functions with only even powers of x.

The Given Function

---

The given function is $f(x) = x^4 - x^3$. To determine whether this function is even, we need to check if $f(-x) = f(x)$.

Option A: Determining Equivalence

---

Option A suggests that we determine whether $(-x)^4 - (-x)^3$ is equivalent to $x^4 - x^3$. Let's evaluate this option.

Expanding $(-x)^4$ and $(-x)^3$

---

When we expand $(-x)^4$, we get $(-x)^4 = x^4$ because the power of 4 is even, and the negative sign is eliminated. Similarly, when we expand $(-x)^3$, we get $(-x)^3 = -x^3$ because the power of 3 is odd, and the negative sign is preserved.

Substituting the Expanded Terms

---

Now, let's substitute the expanded terms into the expression $(-x)^4 - (-x)^3$. We get:

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

Simplifying the Expression

---

When we simplify the expression, we get:

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

Comparing the Results

---

Now, let's compare the result we obtained in Option A with the original function $f(x) = x^4 - x^3$. We can see that the two expressions are not equivalent.

Conclusion

---

Based on the analysis, we can conclude that Option A is not the correct approach to determine whether the given function is even.

Option B: Determining Equivalence

---

Option B suggests that we determine whether $(-x)^4 - (-x)^3$ is equivalent to $f(x) = x^4 - x^3$. Let's evaluate this option.

Expanding $(-x)^4$ and $(-x)^3$

---

As we did in Option A, we can expand $(-x)^4$ and $(-x)^3$ as follows:

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

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

Substituting the Expanded Terms

---

Now, let's substitute the expanded terms into the expression $(-x)^4 - (-x)^3$. We get:

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

Simplifying the Expression

---

When we simplify the expression, we get:

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

Comparing the Results

---

Now, let's compare the result we obtained in Option B with the original function $f(x) = x^4 - x^3$. We can see that the two expressions are not equivalent.

Conclusion

---

Based on the analysis, we can conclude that Option B is not the correct approach to determine whether the given function is even.

The Correct Approach

---

To determine whether the given function is even, we need to check if $f(-x) = f(x)$. Let's evaluate this condition.

Evaluating $f(-x)$

---

When we substitute $-x$ into the function $f(x) = x^4 - x^3$, we get:

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

Expanding $(-x)^4$ and $(-x)^3$

---

As we did earlier, we can expand $(-x)^4$ and $(-x)^3$ as follows:

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

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

Substituting the Expanded Terms

---

Now, let's substitute the expanded terms into the expression $f(-x)$. We get:

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

Simplifying the Expression

---

When we simplify the expression, we get:

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

Comparing the Results

---

Now, let's compare the result we obtained in the correct approach with the original function $f(x) = x^4 - x^3$. We can see that the two expressions are not equivalent.

Conclusion

---

Based on the analysis, we can conclude that the given function $f(x) = x^4 - x^3$ is not an even function.

Final Answer

---

The final answer is: A

=============================================

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 output remains the same. Even functions have symmetry about the y-axis, and they are often represented by functions with only even powers of x.

Q&A: Even Functions

---

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.

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

---

A: To determine if a function is even, you need to check if $f(-x) = f(x)$. If the function satisfies this condition, then it is an even function.

Q: What are some examples of even functions?

---

A: Some examples of even functions include:

• $f(x) = x^2$
• $f(x) = x^4$
• $f(x) = \sin^2(x)$

Q: What are some examples of functions that are not even?

---

A: Some examples of functions that are not even include:

• $f(x) = x^3$
• $f(x) = x^2 + x$
• $f(x) = \sin(x)$

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 satisfies the condition $f(-x) = f(x)$, and if it is odd, then it satisfies the condition $f(-x) = -f(x)$.

Q: How do I graph an even function?

---

A: To graph an even function, you can use the following steps:

1. Find the x-intercepts of the function.
2. Find the y-intercept of the function.
3. Plot the x-intercepts and the y-intercept on a coordinate plane.
4. Draw a smooth curve through the points, making sure that the curve is symmetric about the y-axis.

Q: What are some real-world applications of even functions?

---

A: Even functions have many real-world applications, including:

• Physics: Even functions are used to describe the motion of objects in physics, such as the motion of a pendulum.
• Engineering: Even functions are used to design and analyze systems, such as electrical circuits and mechanical systems.
• Computer Science: Even functions are used in algorithms and data structures, such as sorting and searching.

Conclusion

---

In conclusion, even functions are an important concept in mathematics, and they have many real-world applications. By understanding even functions, you can better analyze and solve problems in physics, engineering, and computer science.

Final Answer

---

The final answer is: A