Possible Forms of the Function

Oct 13, 2025 by recoveryshake.com 78 views

Introduction

In mathematics, the x-intercepts of a function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or zeros of the function. Given the x-intercepts of a function, we can use this information to determine the possible forms of the function. In this article, we will explore how to find a function with given x-intercepts.

Understanding X-Intercepts

The x-intercepts of a function are the values of x where the function equals zero. In other words, if a function f(x) has an x-intercept at (a, 0), then f(a) = 0. The x-intercepts are also the solutions to the equation f(x) = 0.

Given X-Intercepts

The given x-intercepts are ${$ \left(\frac{1}{2}, 0\right)$}$ and ${$ (6, 0)$}$. This means that the function g(x) equals zero at these two points.

Possible Forms of the Function

To find a possible form of the function g(x), we can use the fact that the function equals zero at the given x-intercepts. This means that the function can be written in the form:

g(x) = a(x - r1)(x - r2)

where r1 and r2 are the x-intercepts, and a is a constant.

Option A: g(x) = 2(x + 1)(x + 6)

Let's examine option A: g(x) = 2(x + 1)(x + 6). We can expand this expression to get:

g(x) = 2(x^2 + 7x + 6)

This function has an x-intercept at (−1, 0) and (−6, 0), which are not the given x-intercepts.

Option B: g(x) = (x - 6)(2x - 1)

Now, let's examine option B: g(x) = (x - 6)(2x - 1). We can expand this expression to get:

g(x) = 2x^2 - 13x + 6

This function has an x-intercept at ${$ \left(\frac{1}{2}, 0\right)$}$ and ${$ (6, 0)$}$, which are the given x-intercepts.

Conclusion

In conclusion, the possible form of the function g(x) with given x-intercepts at ${$ \left(\frac{1}{2}, 0\right)$}$ and ${$ (6, 0)$}$ is g(x) = (x - 6)(2x - 1).

Final Answer

The final answer is option B: g(x) = (x - 6)(2x - 1).

Additional Information

The x-intercepts of a function are the points where the graph of the function crosses the x-axis.
The x-intercepts are also known as the roots or zeros of the function.
Given the x-intercepts of a function, we can use this information to determine the possible forms of the function.
The function can be written in the form g(x) = a(x - r1)(x - r2), where r1 and r2 are the x-intercepts, and a is a constant.
A Function with Given X-Intercepts: Exploring Possible Functions and Q&A ====================================================================

Introduction

In our previous article, we explored how to find a function with given x-intercepts. We examined two possible forms of the function and determined that the correct answer was option B: g(x) = (x - 6)(2x - 1). In this article, we will continue to explore the topic of functions with given x-intercepts and answer some frequently asked questions.

Q&A

Q: What are x-intercepts?

A: The x-intercepts of a function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or zeros of the function.

Q: How do I find the x-intercepts of a function?

A: To find the x-intercepts of a function, you can set the function equal to zero and solve for x. This will give you the values of x where the function equals zero.

Q: What is the general form of a function with given x-intercepts?

A: The general form of a function with given x-intercepts is g(x) = a(x - r1)(x - r2), where r1 and r2 are the x-intercepts, and a is a constant.

Q: How do I determine the possible forms of a function with given x-intercepts?

A: To determine the possible forms of a function with given x-intercepts, you can use the fact that the function equals zero at the given x-intercepts. This means that the function can be written in the form g(x) = a(x - r1)(x - r2), where r1 and r2 are the x-intercepts, and a is a constant.

Q: What is the difference between the x-intercepts and the roots of a function?

A: The x-intercepts and the roots of a function are the same thing. They are the points where the graph of the function crosses the x-axis.

Q: Can a function have more than two x-intercepts?

A: Yes, a function can have more than two x-intercepts. In this case, the function can be written in the form g(x) = a(x - r1)(x - r2)(x - r3)..., where r1, r2, r3, ... are the x-intercepts, and a is a constant.

Q: How do I find the x-intercepts of a function with more than two x-intercepts?

A: To find the x-intercepts of a function with more than two x-intercepts, you can set the function equal to zero and solve for x. This will give you the values of x where the function equals zero.

Q: What is the significance of the x-intercepts of a function?

A: The x-intercepts of a function are significant because they give us information about the behavior of the function. For example, if a function has an x-intercept at (a, 0), then the function equals zero at x = a.

Conclusion

In conclusion, the x-intercepts of a function are an important concept in mathematics. They give us information about the behavior of the function and can be used to determine the possible forms of the function. We hope that this article has been helpful in answering your questions about functions with given x-intercepts.

Final Answer

The final answer is that the x-intercepts of a function are the points where the graph of the function crosses the x-axis, and they can be used to determine the possible forms of the function.

Additional Information

The x-intercepts of a function are also known as the roots or zeros of the function.
The x-intercepts of a function can be found by setting the function equal to zero and solving for x.
The general form of a function with given x-intercepts is g(x) = a(x - r1)(x - r2), where r1 and r2 are the x-intercepts, and a is a constant.
A function can have more than two x-intercepts, in which case the function can be written in the form g(x) = a(x - r1)(x - r2)(x - r3)..., where r1, r2, r3, ... are the x-intercepts, and a is a constant.