Which Statement Describes The Behavior Of The Function $f(x) = \frac{3x}{4-x}$?A. The Graph Approaches -3 As $x$ Approaches Infinity.B. The Graph Approaches 0 As $x$ Approaches Infinity.C. The Graph Approaches 3 As

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Understanding the Function

The given function is f(x)=3x4−xf(x) = \frac{3x}{4-x}. This is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is 3x3x and the denominator is 4−x4-x. To understand the behavior of this function, we need to analyze its components and how they interact with each other.

Horizontal Asymptotes

One way to analyze the behavior of a rational function is to look for its horizontal asymptotes. A horizontal asymptote is a horizontal line that the graph of the function approaches as xx approaches infinity or negative infinity. To find the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

Degree of the Numerator and Denominator

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the degree of the numerator is 1 (since the highest power of xx is 1), and the degree of the denominator is 1 (since the highest power of xx is 1). Since the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the highest power of the variable. In this case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is -1. Therefore, the horizontal asymptote is given by the ratio of these two coefficients, which is −31=−3-\frac{3}{1} = -3.

Behavior of the Function as xx Approaches Infinity

Since the horizontal asymptote is y=−3y = -3, the graph of the function approaches this line as xx approaches infinity. This means that as xx gets larger and larger, the value of the function gets closer and closer to −3-3.

Behavior of the Function as xx Approaches Negative Infinity

To analyze the behavior of the function as xx approaches negative infinity, we can use a similar approach. As xx gets more and more negative, the value of the function gets closer and closer to −3-3. This is because the denominator 4−x4-x becomes more and more negative, which causes the value of the function to increase.

Conclusion

In conclusion, the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity. This is because the horizontal asymptote of the function is y=−3y = -3, and the graph of the function approaches this line as xx gets larger and larger.

Comparison with Other Options

Now that we have analyzed the behavior of the function, let's compare it with the other options.

Option A: The graph approaches -3 as xx approaches infinity.

This option is correct, as we have shown that the graph of the function approaches the line y=−3y = -3 as xx approaches infinity.

Option B: The graph approaches 0 as xx approaches infinity.

This option is incorrect, as we have shown that the graph of the function approaches the line y=−3y = -3 as xx approaches infinity, not y=0y = 0.

Option C: The graph approaches 3 as xx approaches infinity.

This option is incorrect, as we have shown that the graph of the function approaches the line y=−3y = -3 as xx approaches infinity, not y=3y = 3.

Final Answer

The final answer is that the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity. This is the correct option, and it is supported by our analysis of the function.

Additional Analysis

In addition to analyzing the behavior of the function as xx approaches infinity, we can also analyze its behavior as xx approaches negative infinity. As we have shown, the graph of the function approaches the line y=−3y = -3 as xx gets more and more negative.

Graphical Analysis

To visualize the behavior of the function, we can graph it using a graphing calculator or a computer algebra system. The graph of the function will show that it approaches the line y=−3y = -3 as xx approaches infinity, and it will also show that it approaches the line y=−3y = -3 as xx approaches negative infinity.

Conclusion

In conclusion, the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. This is supported by our analysis of the function, and it is also supported by the graphical analysis of the function.

Final Answer

The final answer is that the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. This is the correct option, and it is supported by our analysis of the function.

Understanding the Function

The given function is f(x)=3x4−xf(x) = \frac{3x}{4-x}. This is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is 3x3x and the denominator is 4−x4-x. To understand the behavior of this function, we need to analyze its components and how they interact with each other.

Q&A

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is given by the ratio of the leading coefficients of the numerator and denominator. In this case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is -1. Therefore, the horizontal asymptote is given by the ratio of these two coefficients, which is −31=−3-\frac{3}{1} = -3.

Q: What happens to the graph of the function as xx approaches infinity?

A: As xx approaches infinity, the graph of the function approaches the horizontal asymptote, which is y=−3y = -3. This means that as xx gets larger and larger, the value of the function gets closer and closer to −3-3.

Q: What happens to the graph of the function as xx approaches negative infinity?

A: As xx approaches negative infinity, the graph of the function approaches the horizontal asymptote, which is y=−3y = -3. This means that as xx gets more and more negative, the value of the function gets closer and closer to −3-3.

Q: Is the function continuous?

A: Yes, the function is continuous. This means that the graph of the function is a single, unbroken curve, and there are no gaps or holes in the graph.

Q: Is the function differentiable?

A: Yes, the function is differentiable. This means that the graph of the function has a well-defined slope at every point, and the function can be differentiated using the standard rules of differentiation.

Q: What is the domain of the function?

A: The domain of the function is all real numbers except x=4x = 4. This means that the function is defined for all values of xx except x=4x = 4, where the denominator is equal to zero.

Q: What is the range of the function?

A: The range of the function is all real numbers except y=3y = 3. This means that the function can take on any value except y=3y = 3, where the function is undefined.

Conclusion

In conclusion, the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. The function is continuous and differentiable, and its domain is all real numbers except x=4x = 4. The range of the function is all real numbers except y=3y = 3.

Final Answer

The final answer is that the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. This is the correct option, and it is supported by our analysis of the function.

Additional Analysis

In addition to analyzing the behavior of the function as xx approaches infinity, we can also analyze its behavior as xx approaches negative infinity. As we have shown, the graph of the function approaches the line y=−3y = -3 as xx gets more and more negative.

Graphical Analysis

To visualize the behavior of the function, we can graph it using a graphing calculator or a computer algebra system. The graph of the function will show that it approaches the line y=−3y = -3 as xx approaches infinity, and it will also show that it approaches the line y=−3y = -3 as xx approaches negative infinity.

Conclusion

In conclusion, the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. This is supported by our analysis of the function, and it is also supported by the graphical analysis of the function.

Final Answer

The final answer is that the graph of the function f(x)=3x4−xf(x) = \frac{3x}{4-x} approaches the line y=−3y = -3 as xx approaches infinity, and it also approaches the line y=−3y = -3 as xx approaches negative infinity. This is the correct option, and it is supported by our analysis of the function.