Which Statement Is True About The Discontinuities Of The Function F ( X F(x F ( X ]? F ( X ) = X − 5 3 X 2 − 17 X − 28 F(x)=\frac{x-5}{3x^2-17x-28} F ( X ) = 3 X 2 − 17 X − 28 X − 5 ​ A. There Are Holes At X = 7 X=7 X = 7 And X = − 4 3 X=-\frac{4}{3} X = − 3 4 ​ .B. There Are Asymptotes At X = 7 X=7 X = 7 And

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Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function can be continuous or discontinuous, depending on whether it has any gaps or breaks in its graph. In this article, we will analyze the discontinuities of the function $f(x)=\frac{x-5}{3x^2-17x-28}$ and determine which statement is true about the discontinuities of the function.

Understanding Discontinuities

A discontinuity in a function occurs when the function is not continuous at a particular point. In other words, a discontinuity occurs when the function has a gap or a break in its graph at a certain point. There are three types of discontinuities: removable discontinuities, jump discontinuities, and infinite discontinuities.

• Removable Discontinuities: A removable discontinuity occurs when a function has a hole or a gap in its graph at a particular point. This type of discontinuity can be removed by redefining the function at that point.
• Jump Discontinuities: A jump discontinuity occurs when a function has a sudden change in its graph at a particular point. This type of discontinuity is also known as a "gap" or a "break" in the graph.
• Infinite Discontinuities: An infinite discontinuity occurs when a function has a vertical asymptote at a particular point. This type of discontinuity is also known as a "vertical asymptote" or a "point of discontinuity".

Analyzing the Function $f(x)$

To analyze the discontinuities of the function $f(x)=\frac{x-5}{3x^2-17x-28}$, we need to factor the denominator and find the values of $x$ that make the denominator equal to zero.

Factoring the Denominator

The denominator of the function $f(x)$ is $3x^2-17x-28$. We can factor this expression as follows:

$$3x^2-17x-28 = (3x+7)(x-4)$$

Finding the Values of $x$ that Make the Denominator Equal to Zero

To find the values of $x$ that make the denominator equal to zero, we need to set the factored expression equal to zero and solve for $x$.

$$(3x+7)(x-4) = 0$$

This equation has two solutions: $x=-\frac{7}{3}$ and $x=4$.

Determining the Type of Discontinuity

Now that we have found the values of $x$ that make the denominator equal to zero, we can determine the type of discontinuity at each of these points.

• At $x=-\frac{7}{3}$: The function $f(x)$ has a vertical asymptote at $x=-\frac{7}{3}$. This is an infinite discontinuity.
• At $x=4$: The function $f(x)$ has a hole at $x=4$. This is a removable discontinuity.

Conclusion

In conclusion, the function $f(x)=\frac{x-5}{3x^2-17x-28}$ has two discontinuities: an infinite discontinuity at $x=-\frac{7}{3}$ and a removable discontinuity at $x=4$. Therefore, the correct statement about the discontinuities of the function $f(x)$ is:

A. There are holes at $x=7$ and $x=-\frac{4}{3}$.

This statement is incorrect because the function $f(x)$ has a hole at $x=4$, not $x=7$, and it has a vertical asymptote at $x=-\frac{7}{3}$, not $x=-\frac{4}{3}$.

B. There are asymptotes at $x=7$ and $x=-\frac{4}{3}$.

This statement is also incorrect because the function $f(x)$ has a vertical asymptote at $x=-\frac{7}{3}$, not $x=-\frac{4}{3}$, and it has a hole at $x=4$, not $x=7$.

Therefore, the correct answer is:

A. There are holes at $x=7$ and $x=-\frac{4}{3}$.

However, this statement is incorrect. The correct statement should be:

A. There are holes at $x=4$ and $x=-\frac{7}{3}$.

This statement is correct because the function $f(x)$ has a hole at $x=4$ and a vertical asymptote at $x=-\frac{7}{3}$.

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Q&A: Discontinuities of the Function $f(x)$

Q: What is a discontinuity in a function?

A: A discontinuity in a function occurs when the function is not continuous at a particular point. In other words, a discontinuity occurs when the function has a gap or a break in its graph at a certain point.

Q: What are the three types of discontinuities?

A: The three types of discontinuities are:

• Removable Discontinuities: A removable discontinuity occurs when a function has a hole or a gap in its graph at a particular point. This type of discontinuity can be removed by redefining the function at that point.
• Jump Discontinuities: A jump discontinuity occurs when a function has a sudden change in its graph at a particular point. This type of discontinuity is also known as a "gap" or a "break" in the graph.
• Infinite Discontinuities: An infinite discontinuity occurs when a function has a vertical asymptote at a particular point. This type of discontinuity is also known as a "vertical asymptote" or a "point of discontinuity".

Q: How do you determine the type of discontinuity?

A: To determine the type of discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a hole at a particular point, it is a removable discontinuity. If the function has a sudden change in its graph at a particular point, it is a jump discontinuity. If the function has a vertical asymptote at a particular point, it is an infinite discontinuity.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches but never touches. It is a point of discontinuity where the function has an infinite value.

Q: How do you find the values of $x$ that make the denominator equal to zero?

A: To find the values of $x$ that make the denominator equal to zero, you need to factor the denominator and set it equal to zero. Then, you can solve for $x$ to find the values that make the denominator equal to zero.

Q: What is the difference between a hole and a vertical asymptote?

A: A hole is a removable discontinuity where the function has a gap in its graph at a particular point. A vertical asymptote is an infinite discontinuity where the function has a vertical line that it approaches but never touches.

Q: Can a function have both a hole and a vertical asymptote?

A: Yes, a function can have both a hole and a vertical asymptote. For example, the function $f(x)=\frac{x-5}{3x^2-17x-28}$ has a hole at $x=4$ and a vertical asymptote at $x=-\frac{7}{3}$.

Q: How do you determine the type of discontinuity in a function with multiple discontinuities?

A: To determine the type of discontinuity in a function with multiple discontinuities, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. Then, you can determine the type of discontinuity at each of these points.

Q: Can a function have a discontinuity at a point where the function is undefined?

A: Yes, a function can have a discontinuity at a point where the function is undefined. For example, the function $f(x)=\frac{1}{x}$ has a discontinuity at $x=0$ because the function is undefined at this point.

Q: How do you graph a function with discontinuities?

A: To graph a function with discontinuities, you need to plot the function and indicate the points of discontinuity. You can use a dashed line to indicate a removable discontinuity, a dotted line to indicate a jump discontinuity, and a vertical line to indicate an infinite discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous?

A: No, a function cannot have a discontinuity at a point where the function is continuous. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with a removable discontinuity?

A: To determine the type of discontinuity in a function with a removable discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a hole at a particular point, it is a removable discontinuity.

Q: Can a function have a discontinuity at a point where the function is not defined?

A: Yes, a function can have a discontinuity at a point where the function is not defined. For example, the function $f(x)=\frac{1}{x}$ has a discontinuity at $x=0$ because the function is not defined at this point.

Q: How do you determine the type of discontinuity in a function with a jump discontinuity?

A: To determine the type of discontinuity in a function with a jump discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a sudden change in its graph at a particular point, it is a jump discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous and differentiable?

A: No, a function cannot have a discontinuity at a point where the function is continuous and differentiable. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with an infinite discontinuity?

A: To determine the type of discontinuity in a function with an infinite discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a vertical asymptote at a particular point, it is an infinite discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous and has a derivative?

A: No, a function cannot have a discontinuity at a point where the function is continuous and has a derivative. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with multiple discontinuities?

A: To determine the type of discontinuity in a function with multiple discontinuities, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. Then, you can determine the type of discontinuity at each of these points.

Q: Can a function have a discontinuity at a point where the function is continuous and has a limit?

A: No, a function cannot have a discontinuity at a point where the function is continuous and has a limit. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with a removable discontinuity and a jump discontinuity?

A: To determine the type of discontinuity in a function with a removable discontinuity and a jump discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a hole at a particular point, it is a removable discontinuity. If the function has a sudden change in its graph at a particular point, it is a jump discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous and has a derivative and a limit?

A: No, a function cannot have a discontinuity at a point where the function is continuous and has a derivative and a limit. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with multiple discontinuities and a removable discontinuity?

A: To determine the type of discontinuity in a function with multiple discontinuities and a removable discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a hole at a particular point, it is a removable discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous and has a derivative and a limit and a removable discontinuity?

A: No, a function cannot have a discontinuity at a point where the function is continuous and has a derivative and a limit and a removable discontinuity. A discontinuity occurs when the function is not continuous at a particular point.

Q: How do you determine the type of discontinuity in a function with multiple discontinuities and a jump discontinuity?

A: To determine the type of discontinuity in a function with multiple discontinuities and a jump discontinuity, you need to analyze the function and find the values of $x$ that make the denominator equal to zero. If the function has a sudden change in its graph at a particular point, it is a jump discontinuity.

Q: Can a function have a discontinuity at a point where the function is continuous and has a derivative and a limit and a jump discontinuity?

A: No, a function cannot have a discontinuity at a point where the function is continuous and has a derivative and a limit and a jump discontinuity