List The Domain And The \[$ X \$\]- And \[$ Y \$\]-intercepts Of The Following Function. Graph The Function. Be Sure To Label All The Asymptotes.$\[ F(x) = \frac{x^2 + 4x - 5}{x^2 + 6x + 5} \\]

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Introduction

In this article, we will explore the concept of domain and intercepts of a rational function. A rational function is a function that can be expressed as the ratio of two polynomials. The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. The intercepts of a rational function are the points where the graph of the function intersects the x-axis and y-axis.

Domain of a Rational Function

The domain of a rational function is the set of all real numbers except for the values that make the denominator equal to zero. In other words, the domain of a rational function is the set of all real numbers except for the values that make the denominator undefined.

To find the domain of a rational function, we need to find the values of x that make the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for x.

Finding the Domain

Let's consider the rational function:

f(x)=x2+4x−5x2+6x+5{ f(x) = \frac{x^2 + 4x - 5}{x^2 + 6x + 5} }

To find the domain of this function, we need to find the values of x that make the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for x.

x2+6x+5=0{ x^2 + 6x + 5 = 0 }

We can solve this quadratic equation by factoring or using the quadratic formula.

(x+5)(x+1)=0{ (x + 5)(x + 1) = 0 }

This gives us two possible values for x:

x+5=0orx+1=0{ x + 5 = 0 \quad \text{or} \quad x + 1 = 0 }

Solving for x, we get:

x=−5orx=−1{ x = -5 \quad \text{or} \quad x = -1 }

Therefore, the domain of the rational function is all real numbers except for x = -5 and x = -1.

Intercepts of a Rational Function

The intercepts of a rational function are the points where the graph of the function intersects the x-axis and y-axis. To find the x-intercepts of a rational function, we need to find the values of x that make the function equal to zero. To find the y-intercepts of a rational function, we need to find the value of the function when x = 0.

Finding the X-Intercepts

To find the x-intercepts of the rational function, we need to find the values of x that make the function equal to zero. We can do this by setting the numerator equal to zero and solving for x.

x2+4x−5=0{ x^2 + 4x - 5 = 0 }

We can solve this quadratic equation by factoring or using the quadratic formula.

(x+5)(x−1)=0{ (x + 5)(x - 1) = 0 }

This gives us two possible values for x:

x+5=0orx−1=0{ x + 5 = 0 \quad \text{or} \quad x - 1 = 0 }

Solving for x, we get:

x=−5orx=1{ x = -5 \quad \text{or} \quad x = 1 }

Therefore, the x-intercepts of the rational function are x = -5 and x = 1.

Finding the Y-Intercepts

To find the y-intercepts of the rational function, we need to find the value of the function when x = 0. We can do this by substituting x = 0 into the function.

f(0)=(0)2+4(0)−5(0)2+6(0)+5{ f(0) = \frac{(0)^2 + 4(0) - 5}{(0)^2 + 6(0) + 5} }

Simplifying, we get:

f(0)=−55{ f(0) = \frac{-5}{5} }

Therefore, the y-intercept of the rational function is y = -1.

Graphing the Function

To graph the rational function, we need to plot the x-intercepts and y-intercepts on a coordinate plane. We can also use the fact that the graph of a rational function has vertical asymptotes at the values of x that make the denominator equal to zero.

Graphing the Vertical Asymptotes

The vertical asymptotes of the rational function are the lines x = -5 and x = -1. We can plot these lines on a coordinate plane and use them to help us graph the function.

Graphing the Function

Using the x-intercepts, y-intercept, and vertical asymptotes, we can graph the rational function.

Conclusion

In this article, we have explored the concept of domain and intercepts of a rational function. We have found the domain of the rational function by finding the values of x that make the denominator equal to zero. We have also found the x-intercepts and y-intercept of the rational function by setting the numerator equal to zero and substituting x = 0 into the function. Finally, we have graphed the rational function using the x-intercepts, y-intercept, and vertical asymptotes.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Domain and Range of Rational Functions" by Purplemath
  • [3] "Graphing Rational Functions" by Mathway

Table of Contents

  1. Introduction
  2. Domain of a Rational Function
  3. Finding the Domain
  4. Intercepts of a Rational Function
  5. Finding the X-Intercepts
  6. Finding the Y-Intercepts
  7. Graphing the Function
  8. Graphing the Vertical Asymptotes
  9. Conclusion
  10. References
  11. Table of Contents
    Q&A: Domain and Intercepts of a Rational Function =====================================================

Introduction

In our previous article, we explored the concept of domain and intercepts of a rational function. We discussed how to find the domain of a rational function by finding the values of x that make the denominator equal to zero. We also discussed how to find the x-intercepts and y-intercept of a rational function by setting the numerator equal to zero and substituting x = 0 into the function. In this article, we will answer some frequently asked questions about domain and intercepts of a rational function.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all real numbers except for the values that make the denominator equal to zero.

Q: How do I find the domain of a rational function?

A: To find the domain of a rational function, you need to find the values of x that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for x.

Q: What are the x-intercepts of a rational function?

A: The x-intercepts of a rational function are the values of x that make the function equal to zero.

Q: How do I find the x-intercepts of a rational function?

A: To find the x-intercepts of a rational function, you need to set the numerator equal to zero and solve for x.

Q: What is the y-intercept of a rational function?

A: The y-intercept of a rational function is the value of the function when x = 0.

Q: How do I find the y-intercept of a rational function?

A: To find the y-intercept of a rational function, you need to substitute x = 0 into the function.

Q: What are the vertical asymptotes of a rational function?

A: The vertical asymptotes of a rational function are the lines that the graph of the function approaches as x approaches the values that make the denominator equal to zero.

Q: How do I find the vertical asymptotes of a rational function?

A: To find the vertical asymptotes of a rational function, you need to find the values of x that make the denominator equal to zero.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote.

Q: Can a rational function have no vertical asymptotes?

A: Yes, a rational function can have no vertical asymptotes.

Q: How do I graph a rational function?

A: To graph a rational function, you need to plot the x-intercepts, y-intercept, and vertical asymptotes on a coordinate plane.

Q: What are some common mistakes to avoid when graphing a rational function?

A: Some common mistakes to avoid when graphing a rational function include:

  • Not plotting the x-intercepts and y-intercept correctly
  • Not plotting the vertical asymptotes correctly
  • Not using a coordinate plane to graph the function

Conclusion

In this article, we have answered some frequently asked questions about domain and intercepts of a rational function. We have discussed how to find the domain, x-intercepts, y-intercept, and vertical asymptotes of a rational function. We have also discussed how to graph a rational function and some common mistakes to avoid when graphing a rational function.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Domain and Range of Rational Functions" by Purplemath
  • [3] "Graphing Rational Functions" by Mathway

Table of Contents

  1. Introduction
  2. Q: What is the domain of a rational function?
  3. Q: How do I find the domain of a rational function?
  4. Q: What are the x-intercepts of a rational function?
  5. Q: How do I find the x-intercepts of a rational function?
  6. Q: What is the y-intercept of a rational function?
  7. Q: How do I find the y-intercept of a rational function?
  8. Q: What are the vertical asymptotes of a rational function?
  9. Q: Can a rational function have more than one vertical asymptote?
  10. Q: Can a rational function have no vertical asymptotes?
  11. Q: How do I graph a rational function?
  12. Q: What are some common mistakes to avoid when graphing a rational function?
  13. Conclusion
  14. References
  15. Table of Contents