Equations Representing Function m

Oct 13, 2025 by magicalceylontours.com 78 views

Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. They are commonly used in mathematics to model real-world phenomena, such as the behavior of electrical circuits or the growth of populations. In this article, we will explore the concept of rational functions and their vertical asymptotes, and we will use this knowledge to determine which equation could represent a function with the given properties.

What are Rational Functions?

A rational function is a function that can be expressed as the ratio of two polynomials. It is typically written in the form:

f(x) = \frac{p(x)}{q(x)}

where $p(x)$ and $q(x)$ are polynomials. Rational functions can have various properties, such as vertical asymptotes, horizontal asymptotes, and x-intercepts.

Vertical Asymptotes

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of rational functions, vertical asymptotes occur when the denominator of the function is equal to zero. This is because division by zero is undefined, so the function will approach infinity as the denominator approaches zero.

Single x-Intercept at x=5

An x-intercept is a point on the graph of a function where the function crosses the x-axis. In this case, we are given that both functions $m$ and $n$ have a single x-intercept at $x=5$ . This means that both functions will cross the x-axis at the point $(5,0)$ .

Equations Representing Function m

Given that the graphs of rational functions $m$ and $n$ have the same vertical asymptotes and both have a single x-intercept at $x=5$ , we can determine which equation could represent function $m$ . Let's consider the following equation:

m(x) = \frac{x-5}{x^2-4x+3}

This equation has a single x-intercept at $x=5$ and a denominator that is a quadratic polynomial. The denominator can be factored as $(x-3)(x-1)$ , which means that the function will have vertical asymptotes at $x=3$ and $x=1$ .

Equations Representing Function n

Now, let's consider the following equation:

n(x) = \frac{x-5}{x^2-4x+4}

This equation also has a single x-intercept at $x=5$ and a denominator that is a quadratic polynomial. However, the denominator can be factored as $(x-2)^2$ , which means that the function will have a repeated vertical asymptote at $x=2$ .

Which Equation Could Represent Function m?

Based on the given information, we can conclude that the equation that could represent function $m$ is:

m(x) = \frac{x-5}{x^2-4x+3}

This equation has a single x-intercept at $x=5$ and a denominator that is a quadratic polynomial with roots at $x=3$ and $x=1$ , which are the same vertical asymptotes as function $n$ .

Conclusion

In conclusion, we have used the concept of rational functions and their vertical asymptotes to determine which equation could represent function $m$ . We have shown that the equation $m(x) = \frac{x-5}{x^2-4x+3}$ has a single x-intercept at $x=5$ and a denominator that is a quadratic polynomial with roots at $x=3$ and $x=1$ , which are the same vertical asymptotes as function $n$ . This equation could represent function $m$ .

References

[1] "Rational Functions" by Math Open Reference
[2] "Vertical Asymptotes" by Khan Academy
[3] "Single x-Intercept" by Wolfram Alpha

Additional Resources

[1] "Rational Functions" by MIT OpenCourseWare
[2] "Vertical Asymptotes" by University of California, Berkeley
[3] "Single x-Intercept" by University of Michigan
Q&A: Rational Functions and Vertical Asymptotes =====================================================

Frequently Asked Questions

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. It is typically written in the form:

f(x) = \frac{p(x)}{q(x)}

where $p(x)$ and $q(x)$ are polynomials.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of rational functions, vertical asymptotes occur when the denominator of the function is 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. This can be done by factoring the denominator or by using the quadratic formula.

Q: What is a single x-intercept?

A: An x-intercept is a point on the graph of a function where the function crosses the x-axis. In this case, we are given that both functions $m$ and $n$ have a single x-intercept at $x=5$ . This means that both functions will cross the x-axis at the point $(5,0)$ .

Q: How do I determine which equation could represent function m?

A: To determine which equation could represent function m, you need to consider the properties of the function, such as its vertical asymptotes and x-intercepts. In this case, we are given that the graphs of rational functions $m$ and $n$ have the same vertical asymptotes and both have a single x-intercept at $x=5$ . We can use this information to determine which equation could represent function m.

Q: What is the equation that could represent function m?

A: Based on the given information, we can conclude that the equation that could represent function m is:

m(x) = \frac{x-5}{x^2-4x+3}

This equation has a single x-intercept at $x=5$ and a denominator that is a quadratic polynomial with roots at $x=3$ and $x=1$ , which are the same vertical asymptotes as function $n$ .

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

Not factoring the denominator correctly
Not identifying the vertical asymptotes correctly
Not considering the properties of the function, such as its x-intercepts and vertical asymptotes

Q: How can I practice working with rational functions?

A: You can practice working with rational functions by:

Solving problems that involve rational functions
Graphing rational functions
Identifying the vertical asymptotes and x-intercepts of rational functions

Conclusion

In conclusion, we have answered some frequently asked questions about rational functions and vertical asymptotes. We have discussed the properties of rational functions, such as their vertical asymptotes and x-intercepts, and we have used this knowledge to determine which equation could represent function m. We hope that this article has been helpful in understanding rational functions and vertical asymptotes.

References

[1] "Rational Functions" by Math Open Reference
[2] "Vertical Asymptotes" by Khan Academy
[3] "Single x-Intercept" by Wolfram Alpha

Additional Resources

[1] "Rational Functions" by MIT OpenCourseWare
[2] "Vertical Asymptotes" by University of California, Berkeley
[3] "Single x-Intercept" by University of Michigan