Find All Holes Of The Following Function. If A Hole Exists, Write Your Answer As A Coordinate Point In Simplest Form.$\[ F(x) = \frac{5x^2 - 54x + 81}{x^2 - X - 72} \\]Answer Attempt: 1 Out Of 2Options:- Add A Hole- No Holes- [Blank For Answer

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Introduction

Rational functions are a type of mathematical function that involves the division of two polynomials. These functions can have various characteristics, including asymptotes, holes, and vertical tangents. In this article, we will focus on finding holes in rational functions, which are points where the function is not defined due to a zero denominator. We will use the given function as an example to demonstrate the process of finding holes.

Understanding Holes in Rational Functions

A hole in a rational function occurs when there is a zero in the denominator that is also a zero in the numerator. This creates a point where the function is not defined, but the limit of the function as it approaches that point exists. In other words, a hole is a removable discontinuity.

The Given Function

The given function is:

f(x)=5x2−54x+81x2−x−72{ f(x) = \frac{5x^2 - 54x + 81}{x^2 - x - 72} }

To find the holes in this function, we need to factor the numerator and denominator and then identify any common factors.

Factoring the Numerator and Denominator

Let's start by factoring the numerator and denominator:

Factoring the Numerator

The numerator is a quadratic expression that can be factored as follows:

5x2−54x+81=(5x−9)(x−9){ 5x^2 - 54x + 81 = (5x - 9)(x - 9) }

Factoring the Denominator

The denominator is also a quadratic expression that can be factored as follows:

x2−x−72=(x−9)(x+8){ x^2 - x - 72 = (x - 9)(x + 8) }

Identifying Common Factors

Now that we have factored the numerator and denominator, we can identify any common factors. In this case, we can see that both the numerator and denominator have a common factor of (x - 9).

Finding the Hole

Since there is a common factor of (x - 9) in both the numerator and denominator, we can conclude that there is a hole in the function at x = 9. To find the y-coordinate of the hole, we can substitute x = 9 into the function:

f(9)=(5(9)−9)(9−9)(9−9)(9+8){ f(9) = \frac{(5(9) - 9)(9 - 9)}{(9 - 9)(9 + 8)} }

f(9)=(45−9)(0)(0)(17){ f(9) = \frac{(45 - 9)(0)}{(0)(17)} }

f(9)=36(0)0(17){ f(9) = \frac{36(0)}{0(17)} }

f(9)=00{ f(9) = \frac{0}{0} }

This is an indeterminate form, which means that the function is not defined at x = 9. However, we can use the limit to find the value of the function as x approaches 9:

lim⁡x→9f(x)=lim⁡x→9(5x−9)(x−9)(x−9)(x+8){ \lim_{x \to 9} f(x) = \lim_{x \to 9} \frac{(5x - 9)(x - 9)}{(x - 9)(x + 8)} }

lim⁡x→9f(x)=lim⁡x→95x−9x+8{ \lim_{x \to 9} f(x) = \lim_{x \to 9} \frac{5x - 9}{x + 8} }

lim⁡x→9f(x)=5(9)−99+8{ \lim_{x \to 9} f(x) = \frac{5(9) - 9}{9 + 8} }

lim⁡x→9f(x)=45−917{ \lim_{x \to 9} f(x) = \frac{45 - 9}{17} }

lim⁡x→9f(x)=3617{ \lim_{x \to 9} f(x) = \frac{36}{17} }

Therefore, the hole in the function is at the point (9, 36/17).

Conclusion

In this article, we have demonstrated the process of finding holes in rational functions. We used the given function as an example and factored the numerator and denominator to identify any common factors. We then used the limit to find the value of the function as x approaches the point where the hole exists. The hole in the function is at the point (9, 36/17).

Final Answer

The final answer is: (9, 36/17)

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Q: What is a hole in a rational function?

A: A hole in a rational function is a point where the function is not defined due to a zero denominator. However, the limit of the function as it approaches that point exists.

Q: How do I find holes in a rational function?

A: To find holes in a rational function, you need to factor the numerator and denominator and then identify any common factors. If there are common factors, you can conclude that there is a hole in the function at the point where the common factor is zero.

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

A: A hole and a vertical asymptote are both points where the function is not defined. However, a hole is a removable discontinuity, whereas a vertical asymptote is an essential discontinuity. In other words, a hole can be "filled in" by redefining the function at that point, whereas a vertical asymptote cannot be removed.

Q: How do I find the y-coordinate of a hole?

A: To find the y-coordinate of a hole, you can substitute the x-coordinate of the hole into the function. If the function is not defined at that point, you can use the limit to find the value of the function as x approaches that point.

Q: Can a rational function have more than one hole?

A: Yes, a rational function can have more than one hole. This occurs when there are multiple common factors between the numerator and denominator.

Q: How do I determine if a rational function has a hole or a vertical asymptote?

A: To determine if a rational function has a hole or a vertical asymptote, you need to examine the factors of the numerator and denominator. If there are common factors, you can conclude that there is a hole. If there are no common factors, you can conclude that there is a vertical asymptote.

Q: Can a rational function have both holes and vertical asymptotes?

A: Yes, a rational function can have both holes and vertical asymptotes. This occurs when there are multiple common factors between the numerator and denominator, as well as factors that are not common.

Q: How do I graph a rational function with holes and vertical asymptotes?

A: To graph a rational function with holes and vertical asymptotes, you need to identify the points where the function is not defined (holes and vertical asymptotes) and then use the limit to find the value of the function as x approaches those points. You can then plot the points and draw the graph.

Q: What is the significance of holes in rational functions?

A: Holes in rational functions are significant because they indicate points where the function is not defined. However, the limit of the function as it approaches those points exists, which means that the function can be "filled in" at those points.

Q: Can holes in rational functions be removed?

A: Yes, holes in rational functions can be removed by redefining the function at those points. This is done by canceling out the common factors between the numerator and denominator.

Q: How do I remove holes in a rational function?

A: To remove holes in a rational function, you need to cancel out the common factors between the numerator and denominator. This is done by dividing both the numerator and denominator by the common factor.

Q: What is the final answer for the given function?

A: The final answer for the given function is (9, 36/17).