Let $f(x)=\frac{x^2-3x-40}{x^2-x-30}$. Identify The Removable Discontinuity.Removable Discontinuity: $x=$ $\square$

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Introduction

In mathematics, a removable discontinuity is a point at which a function is not defined, but the function can be made continuous by redefining it at that point. In this article, we will focus on identifying removable discontinuities in rational functions. We will use the given function $f(x)=\frac{x2-3x-40}{x2-x-30}$ as an example to demonstrate the process.

Understanding Removable Discontinuities

A removable discontinuity occurs when a rational function has a factor in the numerator that cancels out a corresponding factor in the denominator. This cancellation results in a hole in the graph of the function at the point where the discontinuity occurs.

Step 1: Factor the Numerator and Denominator

To identify the removable discontinuity, we need to factor the numerator and denominator of the given function.

f(x)=x2−3x−40x2−x−30f(x)=\frac{x^2-3x-40}{x^2-x-30}

We can factor the numerator as:

x2−3x−40=(x−8)(x+5)x^2-3x-40 = (x-8)(x+5)

And the denominator as:

x2−x−30=(x−6)(x+5)x^2-x-30 = (x-6)(x+5)

Step 2: Identify the Common Factor

Now that we have factored the numerator and denominator, we can identify the common factor. In this case, the common factor is (x+5)(x+5).

Step 3: Cancel Out the Common Factor

We can cancel out the common factor (x+5)(x+5) from the numerator and denominator.

f(x)=(x−8)(x+5)(x−6)(x+5)f(x)=\frac{(x-8)(x+5)}{(x-6)(x+5)}

f(x)=x−8x−6f(x)=\frac{x-8}{x-6}

Step 4: Identify the Removable Discontinuity

Now that we have cancelled out the common factor, we can identify the removable discontinuity. The removable discontinuity occurs at the point where the cancelled factor is equal to zero.

In this case, the cancelled factor is (x+5)(x+5), which is equal to zero when x=−5x=-5.

Conclusion

In conclusion, we have identified the removable discontinuity in the given rational function $f(x)=\frac{x2-3x-40}{x2-x-30}$. The removable discontinuity occurs at the point x=−5x=-5.

Example Use Case

The concept of removable discontinuity is important in mathematics, particularly in calculus and algebra. It is used to analyze the behavior of functions and to identify points where the function is not defined.

Tips and Tricks

  • To identify removable discontinuities, factor the numerator and denominator of the rational function.
  • Identify the common factor between the numerator and denominator.
  • Cancel out the common factor to simplify the function.
  • Identify the point where the cancelled factor is equal to zero.

Frequently Asked Questions

  • What is a removable discontinuity?
    • A removable discontinuity is a point at which a function is not defined, but the function can be made continuous by redefining it at that point.
  • How do I identify removable discontinuities?
    • Factor the numerator and denominator of the rational function, identify the common factor, cancel out the common factor, and identify the point where the cancelled factor is equal to zero.
  • What is the importance of removable discontinuities?
    • Removable discontinuities are important in mathematics, particularly in calculus and algebra, as they help to analyze the behavior of functions and identify points where the function is not defined.

Conclusion

In conclusion, removable discontinuities are an important concept in mathematics, particularly in calculus and algebra. By following the steps outlined in this article, you can identify removable discontinuities in rational functions and gain a deeper understanding of the behavior of functions.

Removable Discontinuity: x=−5x=\boxed{-5}

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Introduction

In our previous article, we discussed the concept of removable discontinuity in rational functions. In this article, we will provide a comprehensive Q&A section to help you understand the concept better.

Q1: What is a removable discontinuity?

A removable discontinuity is a point at which a function is not defined, but the function can be made continuous by redefining it at that point.

Q2: How do I identify removable discontinuities?

To identify removable discontinuities, follow these steps:

  1. Factor the numerator and denominator of the rational function.
  2. Identify the common factor between the numerator and denominator.
  3. Cancel out the common factor to simplify the function.
  4. Identify the point where the cancelled factor is equal to zero.

Q3: What is the importance of removable discontinuities?

Removable discontinuities are important in mathematics, particularly in calculus and algebra, as they help to analyze the behavior of functions and identify points where the function is not defined.

Q4: Can you provide an example of a removable discontinuity?

Let's consider the function $f(x)=\frac{x2-3x-40}{x2-x-30}$. We can factor the numerator and denominator as:

x2−3x−40=(x−8)(x+5)x^2-3x-40 = (x-8)(x+5)

x2−x−30=(x−6)(x+5)x^2-x-30 = (x-6)(x+5)

The common factor is (x+5)(x+5), which we can cancel out to simplify the function:

f(x)=(x−8)(x+5)(x−6)(x+5)f(x)=\frac{(x-8)(x+5)}{(x-6)(x+5)}

f(x)=x−8x−6f(x)=\frac{x-8}{x-6}

The removable discontinuity occurs at the point where the cancelled factor is equal to zero, which is x=−5x=-5.

Q5: How do I determine if a function has a removable discontinuity?

To determine if a function has a removable discontinuity, follow these steps:

  1. Factor the numerator and denominator of the rational function.
  2. Check if there is a common factor between the numerator and denominator.
  3. If there is a common factor, cancel it out to simplify the function.
  4. If the simplified function has a point where the cancelled factor is equal to zero, then the original function has a removable discontinuity at that point.

Q6: Can a removable discontinuity occur at a point where the function is undefined?

Yes, a removable discontinuity can occur at a point where the function is undefined. In fact, this is the definition of a removable discontinuity.

Q7: How do I handle removable discontinuities in calculus?

In calculus, removable discontinuities are handled by redefining the function at the point of discontinuity. This means that the function is made continuous at the point of discontinuity by redefining it.

Q8: Can a removable discontinuity occur at a point where the function is not differentiable?

Yes, a removable discontinuity can occur at a point where the function is not differentiable. In fact, this is a common scenario in calculus.

Conclusion

In conclusion, removable discontinuities are an important concept in mathematics, particularly in calculus and algebra. By following the steps outlined in this article, you can identify removable discontinuities in rational functions and gain a deeper understanding of the behavior of functions.

Removable Discontinuity: x=−5x=\boxed{-5}