Determine Whether The Function $f(x) = X^2 + X$ Is An Even Function.A. True B. False

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Introduction

In mathematics, an even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. In other words, if a function is even, then its graph is symmetric with respect to the y-axis. In this article, we will determine whether the function f(x) = x^2 + x is an even function.

What is an Even Function?

An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if we replace x with -x in the function, the function remains the same. For example, the function f(x) = x^2 is an even function because f(-x) = (-x)^2 = x^2.

The Function f(x) = x^2 + x

The function f(x) = x^2 + x is a quadratic function. To determine whether it is an even function, we need to check if it satisfies the condition f(-x) = f(x) for all x in its domain.

Checking if f(x) = x^2 + x is an Even Function

To check if f(x) = x^2 + x is an even function, we need to substitute -x for x in the function and see if it remains the same.

f(-x) = (-x)^2 + (-x) f(-x) = x^2 - x

Now, we need to compare f(-x) with f(x).

f(x) = x^2 + x

We can see that f(-x) ≠ f(x) because x^2 - x ≠ x^2 + x. Therefore, the function f(x) = x^2 + x is not an even function.

Conclusion

In conclusion, the function f(x) = x^2 + x is not an even function because it does not satisfy the condition f(-x) = f(x) for all x in its domain. This means that its graph is not symmetric with respect to the y-axis.

Why is it Important to Determine if a Function is Even?

Determining if a function is even is important in mathematics because it helps us understand the properties of the function. For example, if a function is even, then its graph is symmetric with respect to the y-axis, which can help us identify the function's behavior.

Real-World Applications of Even Functions

Even functions have many real-world applications. For example, the function f(x) = x^2 is an even function that models the motion of a pendulum. The function f(x) = x^4 is an even function that models the behavior of a spring.

Common Mistakes to Avoid When Determining if a Function is Even

When determining if a function is even, there are several common mistakes to avoid. For example, some people may substitute -x for x in the function and then simplify the expression without checking if it satisfies the condition f(-x) = f(x). This can lead to incorrect conclusions.

Tips for Determining if a Function is Even

To determine if a function is even, follow these tips:

  • Substitute -x for x in the function and simplify the expression.
  • Check if the simplified expression satisfies the condition f(-x) = f(x).
  • Use the definition of an even function to guide your analysis.

Conclusion

In conclusion, determining if a function is even is an important concept in mathematics. By understanding the properties of even functions, we can better analyze and solve problems involving functions. Remember to follow the tips outlined in this article to ensure accurate results.

Final Answer

The final answer is B. False. The function f(x) = x^2 + x is not an even function because it does not satisfy the condition f(-x) = f(x) for all x in its domain.

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Q1: What is an even function?

An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if we replace x with -x in the function, the function remains the same.

Q2: How do I determine if a function is even?

To determine if a function is even, follow these steps:

  1. Substitute -x for x in the function and simplify the expression.
  2. Check if the simplified expression satisfies the condition f(-x) = f(x).
  3. Use the definition of an even function to guide your analysis.

Q3: What are some common mistakes to avoid when determining if a function is even?

Some common mistakes to avoid when determining if a function is even include:

  • Substituting -x for x in the function and then simplifying the expression without checking if it satisfies the condition f(-x) = f(x).
  • Not using the definition of an even function to guide your analysis.
  • Not checking if the function is defined for all x in its domain.

Q4: What are some real-world applications of even functions?

Even functions have many real-world applications, including:

  • Modeling the motion of a pendulum using the function f(x) = x^2.
  • Modeling the behavior of a spring using the function f(x) = x^4.
  • Modeling the growth of a population using the function f(x) = x^3.

Q5: Can a function be both even and odd?

No, a function cannot be both even and odd. If a function is even, then it satisfies the condition f(-x) = f(x) for all x in its domain. If a function is odd, then it satisfies the condition f(-x) = -f(x) for all x in its domain. These two conditions are mutually exclusive.

Q6: How do I determine if a function is odd?

To determine if a function is odd, follow these steps:

  1. Substitute -x for x in the function and simplify the expression.
  2. Check if the simplified expression satisfies the condition f(-x) = -f(x).
  3. Use the definition of an odd function to guide your analysis.

Q7: What are some common mistakes to avoid when determining if a function is odd?

Some common mistakes to avoid when determining if a function is odd include:

  • Substituting -x for x in the function and then simplifying the expression without checking if it satisfies the condition f(-x) = -f(x).
  • Not using the definition of an odd function to guide your analysis.
  • Not checking if the function is defined for all x in its domain.

Q8: Can a function be neither even nor odd?

Yes, a function can be neither even nor odd. For example, the function f(x) = x^3 is neither even nor odd because it does not satisfy either of the conditions f(-x) = f(x) or f(-x) = -f(x) for all x in its domain.

Q9: How do I determine if a function is neither even nor odd?

To determine if a function is neither even nor odd, follow these steps:

  1. Substitute -x for x in the function and simplify the expression.
  2. Check if the simplified expression satisfies neither of the conditions f(-x) = f(x) or f(-x) = -f(x).
  3. Use the definition of even and odd functions to guide your analysis.

Q10: What are some real-world applications of functions that are neither even nor odd?

Functions that are neither even nor odd have many real-world applications, including:

  • Modeling the growth of a population using the function f(x) = x^3.
  • Modeling the behavior of a complex system using the function f(x) = x^2 + x.
  • Modeling the motion of an object using the function f(x) = x^4 + x^2.

Conclusion

In conclusion, determining if a function is even or odd is an important concept in mathematics. By understanding the properties of even and odd functions, we can better analyze and solve problems involving functions. Remember to follow the tips outlined in this article to ensure accurate results.