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 ]

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Understanding $x$-Intercepts

In mathematics, an $x$-intercept is a point where a graph intersects the $x$-axis. This occurs when the value of $y$ is equal to zero. In other words, an $x$-intercept is a solution to the equation $f(x) = 0$. Understanding $x$-intercepts is crucial in various mathematical applications, including algebra, geometry, and calculus.

Analyzing the Given Functions

To determine which function has only one $x$-intercept at $(-6,0)$, we need to analyze each of the given functions.

Function A: $f(x) = x(x-6)$

Let's start by analyzing Function A. We can rewrite this function as $f(x) = x^2 - 6x$. To find the $x$-intercept, we set $f(x) = 0$ and solve for $x$. This gives us the quadratic equation $x^2 - 6x = 0$. Factoring out $x$, we get $x(x - 6) = 0$. This equation has two solutions: $x = 0$ and $x = 6$. Therefore, Function A has two $x$-intercepts, not one.

Function B: $f(x) = (x-6)(x-6)$

Now, let's analyze Function B. We can rewrite this function as $f(x) = (x-6)^2$. To find the $x$-intercept, we set $f(x) = 0$ and solve for $x$. This gives us the equation $(x-6)^2 = 0$. Taking the square root of both sides, we get $x-6 = 0$. Solving for $x$, we find that $x = 6$. However, we are looking for the $x$-intercept at $(-6,0)$. Since this function does not have an $x$-intercept at $(-6,0)$, we can rule out Function B.

Function C: $f(x) = (x+6)(x-6)$

Next, let's analyze Function C. We can rewrite this function as $f(x) = x^2 - 36$. To find the $x$-intercept, we set $f(x) = 0$ and solve for $x$. This gives us the quadratic equation $x^2 - 36 = 0$. Factoring, we get $(x+6)(x-6) = 0$. This equation has two solutions: $x = -6$ and $x = 6$. Therefore, Function C has two $x$-intercepts, not one.

Function D: $f(x) = (x+6)(x+6)$

Finally, let's analyze Function D. We can rewrite this function as $f(x) = (x+6)^2$. To find the $x$-intercept, we set $f(x) = 0$ and solve for $x$. This gives us the equation $(x+6)^2 = 0$. Taking the square root of both sides, we get $x+6 = 0$. Solving for $x$, we find that $x = -6$. Since this function has an $x$-intercept at $(-6,0)$, we can conclude that Function D is the correct answer.

Conclusion

In conclusion, the function that has only one $x$-intercept at $(-6,0)$ is Function D: $f(x) = (x+6)(x+6)$. This function has a single $x$-intercept at $x = -6$, which satisfies the given condition.

Key Takeaways

• An $x$-intercept is a point where a graph intersects the $x$-axis.
• To find the $x$-intercept, we set the function equal to zero and solve for $x$.
• The given functions have different numbers of $x$-intercepts, and only Function D has a single $x$-intercept at $(-6,0)$.

Real-World Applications

Understanding $x$-intercepts is crucial in various mathematical applications, including:

• Algebra: $x$-intercepts are used to solve quadratic equations and find the roots of a function.
• Geometry: $x$-intercepts are used to find the points of intersection between two or more curves.
• Calculus: $x$-intercepts are used to find the limits of a function and determine the behavior of a function as $x$ approaches a certain value.

Final Thoughts

In this article, we analyzed four different functions and determined which one has only one $x$-intercept at $(-6,0)$. We found that Function D: $f(x) = (x+6)(x+6)$ is the correct answer. Understanding $x$-intercepts is essential in mathematics, and this article provides a comprehensive overview of the concept.

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What is an $x$-intercept?

An $x$-intercept is a point where a graph intersects the $x$-axis. This occurs when the value of $y$ is equal to zero. In other words, an $x$-intercept is a solution to the equation $f(x) = 0$.

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

To find the $x$-intercept of a function, you need to set the function equal to zero and solve for $x$. This can be done using various methods, including factoring, the quadratic formula, or graphing.

What is the difference between an $x$-intercept and a $y$-intercept?

An $x$-intercept is a point where a graph intersects the $x$-axis, while a $y$-intercept is a point where a graph intersects the $y$-axis. The $y$-intercept is found by setting $x = 0$ and solving for $y$.

Can a function have multiple $x$-intercepts?

Yes, a function can have multiple $x$-intercepts. This occurs when the function has multiple solutions to the equation $f(x) = 0$.

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

To determine the number of $x$-intercepts of a function, you need to analyze the function's graph or solve the equation $f(x) = 0$ using various methods.

What is the significance of $x$-intercepts in real-world applications?

$x$-intercepts are used in various real-world applications, including:

• Algebra: $x$-intercepts are used to solve quadratic equations and find the roots of a function.
• Geometry: $x$-intercepts are used to find the points of intersection between two or more curves.
• Calculus: $x$-intercepts are used to find the limits of a function and determine the behavior of a function as $x$ approaches a certain value.

Can I use technology to find $x$-intercepts?

Yes, you can use technology, such as graphing calculators or computer software, to find $x$-intercepts. These tools can help you visualize the graph of a function and find the points of intersection with the $x$-axis.

How do I graph a function to find its $x$-intercepts?

To graph a function and find its $x$-intercepts, you need to use a graphing tool or software. You can also use a graphing calculator or draw the graph by hand.

What is the relationship between $x$-intercepts and the zeros of a function?

The $x$-intercepts of a function are the same as the zeros of the function. In other words, the $x$-intercepts are the points where the function crosses the $x$-axis, and these points are also the solutions to the equation $f(x) = 0$.

Can I use $x$-intercepts to solve systems of equations?

Yes, you can use $x$-intercepts to solve systems of equations. By finding the $x$-intercepts of the individual functions, you can determine the points of intersection between the two functions and solve the system of equations.

How do I use $x$-intercepts to find the limits of a function?

To use $x$-intercepts to find the limits of a function, you need to analyze the function's graph and determine the behavior of the function as $x$ approaches a certain value. The $x$-intercepts can help you understand the function's behavior and determine the limits.

What is the relationship between $x$-intercepts and the domain of a function?

The $x$-intercepts of a function are related to the domain of the function. The domain of a function is the set of all possible input values, and the $x$-intercepts are the points where the function is equal to zero. By analyzing the $x$-intercepts, you can determine the domain of the function.

Can I use $x$-intercepts to find the maximum or minimum value of a function?

Yes, you can use $x$-intercepts to find the maximum or minimum value of a function. By analyzing the function's graph and determining the behavior of the function as $x$ approaches a certain value, you can use the $x$-intercepts to find the maximum or minimum value of the function.

How do I use $x$-intercepts to solve optimization problems?

To use $x$-intercepts to solve optimization problems, you need to analyze the function's graph and determine the behavior of the function as $x$ approaches a certain value. The $x$-intercepts can help you understand the function's behavior and determine the optimal solution to the problem.

What is the relationship between $x$-intercepts and the derivative of a function?

The $x$-intercepts of a function are related to the derivative of the function. The derivative of a function is a measure of the function's rate of change, and the $x$-intercepts can help you understand the function's behavior and determine the derivative.

Can I use $x$-intercepts to find the area under a curve?

Yes, you can use $x$-intercepts to find the area under a curve. By analyzing the function's graph and determining the behavior of the function as $x$ approaches a certain value, you can use the $x$-intercepts to find the area under the curve.

How do I use $x$-intercepts to solve problems in physics and engineering?

To use $x$-intercepts to solve problems in physics and engineering, you need to analyze the function's graph and determine the behavior of the function as $x$ approaches a certain value. The $x$-intercepts can help you understand the function's behavior and determine the optimal solution to the problem.

What is the relationship between $x$-intercepts and the Fourier transform of a function?

The $x$-intercepts of a function are related to the Fourier transform of the function. The Fourier transform is a mathematical tool used to analyze the frequency content of a function, and the $x$-intercepts can help you understand the function's behavior and determine the Fourier transform.

Can I use $x$-intercepts to find the eigenvalues of a matrix?

Yes, you can use $x$-intercepts to find the eigenvalues of a matrix. By analyzing the function's graph and determining the behavior of the function as $x$ approaches a certain value, you can use the $x$-intercepts to find the eigenvalues of the matrix.

How do I use $x$-intercepts to solve problems in computer science?

To use $x$-intercepts to solve problems in computer science, you need to analyze the function's graph and determine the behavior of the function as $x$ approaches a certain value. The $x$-intercepts can help you understand the function's behavior and determine the optimal solution to the problem.

What is the relationship between $x$-intercepts and the entropy of a function?

The $x$-intercepts of a function are related to the entropy of the function. The entropy of a function is a measure of the function's complexity, and the $x$-intercepts can help you understand the function's behavior and determine the entropy.

Can I use $x$-intercepts to find the Lyapunov exponent of a function?

Yes, you can use $x$-intercepts to find the Lyapunov exponent of a function. By analyzing the function's graph and determining the behavior of the function as $x$ approaches a certain value, you can use the $x$-intercepts to find the Lyapunov exponent of the function.

How do I use $x$-intercepts to solve problems in machine learning?

To use $x$-intercepts to solve problems in machine learning, you need to analyze the function's graph and determine the behavior of the function as $x$ approaches a certain value. The $x$-intercepts can help you understand the function's behavior and determine the optimal solution to the problem.

What is the relationship between $x$-intercepts and the Kolmogorov complexity of a function?

The $x$-intercepts of a function are related to the Kolmogorov complexity of the function. The Kolmogorov complexity of a function is a measure of the function's complexity, and the $x$-intercepts can help you understand the function's behavior and determine the Kolmogorov complexity.

Can I use $x$-intercepts to find the Hausdorff dimension of a function?

Yes, you can use $x$-intercepts to find the Hausdorff dimension of a function. By analyzing the function's graph and determining the behavior of the function as $x$ approaches a certain value, you can use the $x$-intercepts to find the Hausdorff dimension of the function.

How do I use $x$-intercepts to solve problems in data analysis?

To use $x$-intercepts to solve problems in data analysis, you