The Graph Of The Even Function { F(x) $}$ Has Five X-intercepts. If (6, 0) Is One Of The Intercepts, Which Set Of Points Can Be The Other X-intercepts Of The Graph Of { F(x) $}$?A. (-6, 0), (-2, 0), And (0, 0)B. (-6, 0), (-2, 0),

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Introduction

In mathematics, an even function is a function where { f(-x) = f(x) $}$ for all x in the domain of the function. This property implies that the graph of an even function is symmetric with respect to the y-axis. In this article, we will explore the x-intercepts of the graph of an even function and determine which set of points can be the other x-intercepts of the graph of { f(x) $}$ given that (6, 0) is one of the intercepts.

Understanding Even Functions

An even function is a function where { f(-x) = f(x) $}$ for all x in the domain of the function. This property implies that the graph of an even function is symmetric with respect to the y-axis. In other words, if the point (a, b) is on the graph of an even function, then the point (-a, b) is also on the graph.

X-Intercepts of Even Functions

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. In other words, the x-intercepts are the points where the function value is equal to zero. For an even function, the x-intercepts are symmetric with respect to the y-axis.

Given Information

We are given that (6, 0) is one of the x-intercepts of the graph of { f(x) $}$. This means that { f(6) = 0 $}$.

Possible X-Intercepts

Since the graph of the even function is symmetric with respect to the y-axis, the other x-intercepts must be symmetric with respect to the y-axis. This means that if (6, 0) is an x-intercept, then (-6, 0) must also be an x-intercept.

Determining the Other X-Intercepts

We are given two options for the other x-intercepts:

A. (-6, 0), (-2, 0), and (0, 0) B. (-6, 0), (-2, 0)

To determine which set of points can be the other x-intercepts of the graph of { f(x) $}$, we need to consider the symmetry of the graph with respect to the y-axis.

Option A

If the other x-intercepts are (-6, 0), (-2, 0), and (0, 0), then the graph of the function would have three x-intercepts that are symmetric with respect to the y-axis. However, this would imply that the function has a total of five x-intercepts, which is not possible.

Option B

If the other x-intercepts are (-6, 0) and (-2, 0), then the graph of the function would have two x-intercepts that are symmetric with respect to the y-axis. This would imply that the function has a total of three x-intercepts, which is not possible.

Conclusion

Based on the symmetry of the graph with respect to the y-axis, we can conclude that the other x-intercepts of the graph of { f(x) $}$ must be (-6, 0) and (-2, 0). This is because the graph of the even function is symmetric with respect to the y-axis, and the x-intercepts must be symmetric with respect to the y-axis.

Final Answer

The final answer is:

B. (-6, 0), (-2, 0)

References

  • [1] "Even Functions" by Math Open Reference
  • [2] "Symmetry of Graphs" by Khan Academy

Additional Information

  • Even functions have symmetry with respect to the y-axis.
  • The x-intercepts of an even function are symmetric with respect to the y-axis.
  • The graph of an even function can have multiple x-intercepts.
  • The x-intercepts of an even function can be determined using the symmetry of the graph with respect to the y-axis.
    The Graph of the Even Function: Q&A =====================================

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 property implies that the graph of an even function is symmetric with respect to the y-axis.

Q: What is the significance of symmetry in an even function?

A: The symmetry of an even function with respect to the y-axis means that if the point (a, b) is on the graph of the function, then the point (-a, b) is also on the graph. This symmetry is a key characteristic of even functions.

Q: How do you determine the x-intercepts of an even function?

A: The x-intercepts of an even function are the points where the graph of the function crosses the x-axis. In other words, the x-intercepts are the points where the function value is equal to zero. For an even function, the x-intercepts are symmetric with respect to the y-axis.

Q: What is the relationship between the x-intercepts of an even function and its symmetry?

A: The x-intercepts of an even function are symmetric with respect to the y-axis. This means that if (a, 0) is an x-intercept, then (-a, 0) is also an x-intercept.

Q: How do you determine the other x-intercepts of an even function given one x-intercept?

A: To determine the other x-intercepts of an even function given one x-intercept, you need to consider the symmetry of the graph with respect to the y-axis. The other x-intercepts must be symmetric with respect to the y-axis.

Q: What is the significance of the given x-intercept (6, 0) in the problem?

A: The given x-intercept (6, 0) is one of the x-intercepts of the graph of { f(x) $}$. This means that { f(6) = 0 $}$. The symmetry of the graph with respect to the y-axis implies that (-6, 0) is also an x-intercept.

Q: How do you determine the other x-intercepts of the graph of { f(x) $}$ given the x-intercept (6, 0)?

A: To determine the other x-intercepts of the graph of { f(x) $}$ given the x-intercept (6, 0), you need to consider the symmetry of the graph with respect to the y-axis. The other x-intercepts must be symmetric with respect to the y-axis.

Q: What is the final answer to the problem?

A: The final answer is:

B. (-6, 0), (-2, 0)

Q: What are some additional concepts related to even functions?

A: Some additional concepts related to even functions include:

  • Even functions have symmetry with respect to the y-axis.
  • The x-intercepts of an even function are symmetric with respect to the y-axis.
  • The graph of an even function can have multiple x-intercepts.
  • The x-intercepts of an even function can be determined using the symmetry of the graph with respect to the y-axis.

Q: Where can I learn more about even functions?

A: You can learn more about even functions by visiting the following resources:

  • [1] "Even Functions" by Math Open Reference
  • [2] "Symmetry of Graphs" by Khan Academy

Conclusion

In this Q&A article, we have discussed the properties of even functions, including symmetry with respect to the y-axis and the relationship between x-intercepts and symmetry. We have also determined the other x-intercepts of the graph of { f(x) $}$ given the x-intercept (6, 0).