Which Function Has Only One X X X -intercept At ( − 6 , 0 (-6,0 ( − 6 , 0 ]?A. F ( X ) = X ( X − 6 F(x)=x(x-6 F ( X ) = X ( X − 6 ] B. F ( X ) = ( X − 6 ) ( X − 6 F(x)=(x-6)(x-6 F ( X ) = ( X − 6 ) ( X − 6 ] C. F ( X ) = ( X + 6 ) ( X − 6 F(x)=(x+6)(x-6 F ( X ) = ( X + 6 ) ( X − 6 ] D. F ( X ) = ( X + 6 ) ( X + 6 F(x)=(x+6)(x+6 F ( X ) = ( X + 6 ) ( X + 6 ]

Understanding the Concept of $x$-Intercepts

In mathematics, the $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. This occurs when the value of $y$ is equal to zero. In other words, the $x$-intercept is the solution to the equation $f(x) = 0$. In this article, we will explore the concept of $x$-intercepts and determine which function has only one $x$-intercept at $(-6,0)$.

What are $x$-Intercepts?

$x$-intercepts are an essential concept in algebra and graphing. They are used to identify the points at which a function intersects the $x$-axis. To find the $x$-intercept of a function, we need to set the function equal to zero and solve for $x$. This is because the $x$-axis is defined by the equation $y = 0$. Therefore, when a function intersects the $x$-axis, its value is equal to zero.

Types of $x$-Intercepts

$x$-intercepts can be classified into two types: single $x$-intercepts and multiple $x$-intercepts. A single $x$-intercept occurs when a function intersects the $x$-axis at only one point. On the other hand, multiple $x$-intercepts occur when a function intersects the $x$-axis at more than one point.

Analyzing the Given Functions

Now that we have a good understanding of $x$-intercepts, let's analyze the given functions to determine which one has only one $x$-intercept at $(-6,0)$.

A. $f(x)=x(x-6)$

To find the $x$-intercept of this function, we need to set it equal to zero and solve for $x$. This gives us the equation:

$x(x-6) = 0$

Expanding the equation, we get:

$x^2 - 6x = 0$

Factoring out $x$, we get:

$x(x - 6) = 0$

This equation has two solutions: $x = 0$ and $x = 6$. Therefore, the function $f(x) = x(x-6)$ has two $x$-intercepts at $(0,0)$ and $(6,0)$.

B. $f(x)=(x-6)(x-6)$

To find the $x$-intercept of this function, we need to set it equal to zero and solve for $x$. This gives us the equation:

$(x-6)(x-6) = 0$

Expanding the equation, we get:

$x^2 - 12x + 36 = 0$

This equation has two solutions: $x = 6$ and $x = 6$. Therefore, the function $f(x) = (x-6)(x-6)$ has only one $x$-intercept at $(6,0)$.

C. $f(x)=(x+6)(x-6)$

To find the $x$-intercept of this function, we need to set it equal to zero and solve for $x$. This gives us the equation:

$(x+6)(x-6) = 0$

Expanding the equation, we get:

$x^2 - 36 = 0$

This equation has two solutions: $x = 6$ and $x = -6$. Therefore, the function $f(x) = (x+6)(x-6)$ has two $x$-intercepts at $(6,0)$ and $(-6,0)$.

D. $f(x)=(x+6)(x+6)$

To find the $x$-intercept of this function, we need to set it equal to zero and solve for $x$. This gives us the equation:

$(x+6)(x+6) = 0$

Expanding the equation, we get:

$x^2 + 12x + 36 = 0$

This equation has no real solutions. Therefore, the function $f(x) = (x+6)(x+6)$ has no $x$-intercepts.

Conclusion

Based on our analysis, we can conclude that the function $f(x) = (x-6)(x-6)$ has only one $x$-intercept at $(6,0)$. However, we are looking for a function with an $x$-intercept at $(-6,0)$. Therefore, we need to consider the other options.

From our analysis, we can see that the function $f(x) = (x+6)(x-6)$ has two $x$-intercepts at $(6,0)$ and $(-6,0)$. Therefore, this function meets the criteria of having only one $x$-intercept at $(-6,0)$.

Answer

The correct answer is C. $f(x)=(x+6)(x-6)$
Q&A: Understanding $x$-Intercepts

In our previous article, we explored the concept of $x$-intercepts and determined which function has only one $x$-intercept at $(-6,0)$. In this article, we will answer some frequently asked questions about $x$-intercepts to help you better understand this concept.

Q: What is the difference between a single $x$-intercept and multiple $x$-intercepts?

A: A single $x$-intercept occurs when a function intersects the $x$-axis at only one point. On the other hand, multiple $x$-intercepts occur when a function intersects the $x$-axis at more than one point.

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

A: To find the $x$-intercept of a function, you need to set the function equal to zero and solve for $x$. This is because the $x$-axis is defined by the equation $y = 0$. Therefore, when a function intersects the $x$-axis, its value is equal to zero.

Q: What is the significance of $x$-intercepts in mathematics?

A: $x$-intercepts are an essential concept in algebra and graphing. They are used to identify the points at which a function intersects the $x$-axis. This information can be used to analyze the behavior of a function and make predictions about its behavior.

Q: Can a function have no $x$-intercepts?

A: Yes, a function can have no $x$-intercepts. This occurs when the function does not intersect the $x$-axis at any point. For example, the function $f(x) = x^2 + 1$ has no $x$-intercepts because it is always positive.

Q: How do I determine the number of $x$-intercepts of a function?

A: To determine the number of $x$-intercepts of a function, you need to analyze the function's equation. If the equation has multiple solutions, then the function has multiple $x$-intercepts. If the equation has only one solution, then the function has only one $x$-intercept.

Q: Can a function have a negative $x$-intercept?

A: Yes, a function can have a negative $x$-intercept. This occurs when the function intersects the $x$-axis at a point with a negative $x$-coordinate. For example, the function $f(x) = (x+6)(x-6)$ has an $x$-intercept at $(-6,0)$.

Q: How do I graph a function with multiple $x$-intercepts?

A: To graph a function with multiple $x$-intercepts, you need to plot the function's $x$-intercepts on a coordinate plane. Then, you can use the function's equation to determine the points at which the function intersects the $x$-axis.

Q: Can a function have an $x$-intercept at the origin?

A: Yes, a function can have an $x$-intercept at the origin. This occurs when the function intersects the $x$-axis at the point $(0,0)$. For example, the function $f(x) = x(x-6)$ has an $x$-intercept at $(0,0)$.

Conclusion

In this article, we answered some frequently asked questions about $x$-intercepts to help you better understand this concept. We hope that this information has been helpful in your understanding of $x$-intercepts and their significance in mathematics.

Additional Resources

If you are interested in learning more about $x$-intercepts, we recommend the following resources:

• Khan Academy: $x$-Intercepts
• Mathway: $x$-Intercepts
• Wolfram Alpha: $x$-Intercepts

We hope that this information has been helpful in your understanding of $x$-intercepts. If you have any further questions, please don't hesitate to ask.