For The Function $y=\frac{x}{9-x}$:What Is The Range?A. $-\infty \ \textless \ Y \ \textless \ 1$ And $1 \ \textless \ Y \ \textless \ \infty$ B. $-\infty \ \textless \ Y \leq -1$ And $-1 \leq Y \

by ADMIN 209 views

Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. Rational functions, in particular, are a type of function that can be expressed as the ratio of two polynomials. In this article, we will explore the range of the rational function y=x9−xy=\frac{x}{9-x}.

What is a Rational Function?

A rational function is a function that can be expressed as the ratio of two polynomials. It is typically written in the form y=f(x)g(x)y=\frac{f(x)}{g(x)}, where f(x)f(x) and g(x)g(x) are polynomials. Rational functions can have various types of behavior, including asymptotes, holes, and vertical tangents.

The Given Function

The given function is y=x9−xy=\frac{x}{9-x}. This is a rational function with a numerator of xx and a denominator of 9−x9-x. To find the range of this function, we need to consider the possible output values it can produce for the given input values.

Finding the Range

To find the range of the function, we need to consider the possible output values it can produce for the given input values. We can start by finding the domain of the function, which is the set of all possible input values.

Domain of the Function

The domain of the function is all real numbers except for x=9x=9, since this value would make the denominator zero and the function undefined.

Asymptotes

The function has a vertical asymptote at x=9x=9, since this value makes the denominator zero and the function undefined. The vertical asymptote is a vertical line that the function approaches but never touches.

Holes

The function has a hole at x=0x=0, since this value makes the numerator and denominator both zero, resulting in an undefined value.

Range of the Function

To find the range of the function, we need to consider the possible output values it can produce for the given input values. We can start by finding the minimum and maximum values of the function.

Minimum Value

To find the minimum value of the function, we can use the fact that the function has a vertical asymptote at x=9x=9. This means that the function approaches negative infinity as xx approaches 9 from the left.

Maximum Value

To find the maximum value of the function, we can use the fact that the function has a hole at x=0x=0. This means that the function approaches positive infinity as xx approaches 0 from the right.

Conclusion

In conclusion, the range of the function y=x9−xy=\frac{x}{9-x} is −∞ \textless y \textless 1-\infty \ \textless \ y \ \textless \ 1 and 1 \textless y \textless ∞1 \ \textless \ y \ \textless \ \infty. This means that the function can produce any value between negative infinity and 1, and any value between 1 and positive infinity.

Final Answer

Introduction

In our previous article, we explored the range of the rational function y=x9−xy=\frac{x}{9-x}. In this article, we will answer some frequently asked questions about the range of this function.

Q: What is the domain of the function?

A: The domain of the function is all real numbers except for x=9x=9, since this value would make the denominator zero and the function undefined.

Q: What is the vertical asymptote of the function?

A: The vertical asymptote of the function is x=9x=9, since this value makes the denominator zero and the function undefined.

Q: What is the hole in the function?

A: The hole in the function is x=0x=0, since this value makes the numerator and denominator both zero, resulting in an undefined value.

Q: What is the minimum value of the function?

A: The minimum value of the function is negative infinity, since the function approaches negative infinity as xx approaches 9 from the left.

Q: What is the maximum value of the function?

A: The maximum value of the function is positive infinity, since the function approaches positive infinity as xx approaches 0 from the right.

Q: What is the range of the function?

A: The range of the function is −∞ \textless y \textless 1-\infty \ \textless \ y \ \textless \ 1 and 1 \textless y \textless ∞1 \ \textless \ y \ \textless \ \infty, since the function can produce any value between negative infinity and 1, and any value between 1 and positive infinity.

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

A: To find the range of a rational function, you need to consider the possible output values it can produce for the given input values. You can start by finding the domain of the function, and then use the fact that the function has asymptotes and holes to determine the range.

Q: What are some common mistakes to avoid when finding the range of a rational function?

A: Some common mistakes to avoid when finding the range of a rational function include:

  • Not considering the domain of the function
  • Not identifying the asymptotes and holes of the function
  • Not using the fact that the function approaches positive or negative infinity as xx approaches a certain value
  • Not considering the possibility of the function having a range that includes negative infinity or positive infinity

Conclusion

In conclusion, the range of the function y=x9−xy=\frac{x}{9-x} is −∞ \textless y \textless 1-\infty \ \textless \ y \ \textless \ 1 and 1 \textless y \textless ∞1 \ \textless \ y \ \textless \ \infty. We hope this article has helped you understand the range of this function and how to find the range of a rational function in general.

Final Answer

The final answer is A. −∞ \textless y \textless 1-\infty \ \textless \ y \ \textless \ 1 and 1 \textless y \textless ∞1 \ \textless \ y \ \textless \ \infty.