The Graph Of Function F ( X F(x F ( X ] Has Zeros At − 5 , − 2 -5, -2 − 5 , − 2 , And 2 2 2 . Which Of The Following Could Define F ( X F(x F ( X ]?A. F ( X ) = ( X + 5 ) ( X − 2 ) ( X + 2 F(x) = (x + 5)(x - 2)(x + 2 F ( X ) = ( X + 5 ) ( X − 2 ) ( X + 2 ]B. F ( X ) = ( X + 5 ) ( X − 2 ) 2 F(x) = (x + 5)(x - 2)^2 F ( X ) = ( X + 5 ) ( X − 2 ) 2 C. $f(x) = (x - 2)(x + 2)(x +

Introduction

In mathematics, the graph of a function is a visual representation of the relationship between the input and output values of the function. One of the key characteristics of a function's graph is its zeros, which are the points where the function intersects the x-axis. In this article, we will explore the concept of zeros and how they relate to the possible definitions of a function.

Understanding Zeros

A zero of a function is a value of x that makes the function equal to zero. In other words, if f(x) = 0, then x is a zero of the function. The zeros of a function are important because they provide information about the function's behavior and can be used to determine the function's graph.

Given Information

The problem states that the graph of function $f(x)$ has zeros at $-5, -2$, and $2$. This means that when x = -5, x = -2, or x = 2, the function f(x) is equal to zero.

Possible Definitions

We are given three possible definitions of the function f(x):

A. $f(x) = (x + 5)(x - 2)(x + 2)$ B. $f(x) = (x + 5)(x - 2)^2$ C. $f(x) = (x - 2)(x + 2)(x + 5)$

We need to determine which of these definitions could define the function f(x) given the zeros at $-5, -2$, and $2$.

Analyzing Option A

Let's start by analyzing option A: $f(x) = (x + 5)(x - 2)(x + 2)$. This definition implies that the function has zeros at x = -5, x = 2, and x = -2. However, the problem states that the function has zeros at $-5, -2$, and $2$, not at x = 2 and x = -2. Therefore, option A is not a possible definition of the function f(x).

Analyzing Option B

Next, let's analyze option B: $f(x) = (x + 5)(x - 2)^2$. This definition implies that the function has a zero at x = -5 and two zeros at x = 2. However, the problem states that the function has zeros at $-5, -2$, and $2$, not two zeros at x = 2. Therefore, option B is not a possible definition of the function f(x).

Analyzing Option C

Finally, let's analyze option C: $f(x) = (x - 2)(x + 2)(x + 5)$. This definition implies that the function has zeros at x = 2, x = -2, and x = -5. Therefore, option C is a possible definition of the function f(x) given the zeros at $-5, -2$, and $2$.

Conclusion

In conclusion, the possible definition of the function f(x) given the zeros at $-5, -2$, and $2$ is option C: $f(x) = (x - 2)(x + 2)(x + 5)$. This definition implies that the function has zeros at x = 2, x = -2, and x = -5, which matches the given information.

Key Takeaways

• The zeros of a function are the points where the function intersects the x-axis.
• The zeros of a function provide information about the function's behavior and can be used to determine the function's graph.
• When analyzing possible definitions of a function, it's essential to consider the function's zeros and ensure that they match the given information.

Further Reading

For more information on functions and their graphs, we recommend the following resources:

• Khan Academy: Functions and Graphs
• Mathway: Functions and Graphs
• Wolfram Alpha: Functions and Graphs

Introduction

In our previous article, we explored the concept of zeros and how they relate to the possible definitions of a function. We analyzed three possible definitions of the function f(x) and determined that option C: $f(x) = (x - 2)(x + 2)(x + 5)$ is a possible definition of the function f(x) given the zeros at $-5, -2$, and $2$.

Q&A

Q: What are the zeros of a function?

A: The zeros of a function are the points where the function intersects the x-axis. In other words, if f(x) = 0, then x is a zero of the function.

Q: How do the zeros of a function relate to its graph?

A: The zeros of a function provide information about the function's behavior and can be used to determine the function's graph. The graph of a function will intersect the x-axis at the zeros of the function.

Q: What is the difference between a zero and a root of a function?

A: A zero of a function is a value of x that makes the function equal to zero. A root of a function is a value of x that makes the function equal to zero, but it can also be a value of x that makes the function equal to a multiple of zero (e.g. f(x) = 0, f(x) = 2, etc.).

Q: How do you determine the possible definitions of a function given its zeros?

A: To determine the possible definitions of a function given its zeros, you need to consider the factors of the function. The factors of the function will correspond to the zeros of the function. For example, if the function has zeros at x = -5, x = -2, and x = 2, then the possible definitions of the function will be of the form f(x) = (x + 5)(x - 2)(x + 2).

Q: What is the importance of understanding the zeros of a function?

A: Understanding the zeros of a function is important because it provides information about the function's behavior and can be used to determine the function's graph. It also helps to identify the possible definitions of a function given its zeros.

Q: How do you analyze the possible definitions of a function?

A: To analyze the possible definitions of a function, you need to consider the factors of the function and ensure that they match the given information. You also need to check if the function has any additional zeros that are not accounted for by the factors.

Q: What are some common mistakes to avoid when analyzing the possible definitions of a function?

A: Some common mistakes to avoid when analyzing the possible definitions of a function include:

• Not considering the factors of the function
• Not ensuring that the factors match the given information
• Not checking for additional zeros that are not accounted for by the factors
• Not considering the possibility of multiple definitions of the function

Conclusion

In conclusion, understanding the zeros of a function is essential for determining the possible definitions of a function. By considering the factors of the function and ensuring that they match the given information, you can analyze the possible definitions of a function and determine the correct definition.

Key Takeaways

• The zeros of a function are the points where the function intersects the x-axis.
• The zeros of a function provide information about the function's behavior and can be used to determine the function's graph.
• Understanding the zeros of a function is essential for determining the possible definitions of a function.
• To analyze the possible definitions of a function, you need to consider the factors of the function and ensure that they match the given information.

Further Reading

For more information on functions and their graphs, we recommend the following resources:

• Khan Academy: Functions and Graphs
• Mathway: Functions and Graphs
• Wolfram Alpha: Functions and Graphs

By understanding the concept of zeros and how they relate to the possible definitions of a function, you can better analyze and solve problems involving functions and their graphs.