Not considering the sign of the function

Oct 13, 2025 by windailys.com 97 views

Introduction

When it comes to functions, understanding the range is crucial in mathematics. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we will explore the range of the function $y=\frac{-24}{x^2+8}$ and determine which of the given statements correctly describes it.

The Function in Question

The function in question is $y=\frac{-24}{x^2+8}$ . This is a rational function, where the numerator is a constant and the denominator is a quadratic expression. To find the range, we need to consider the behavior of the function as $x$ varies.

Analyzing the Denominator

The denominator of the function is $x^2+8$ . Since $x^2$ is always non-negative (or zero), the minimum value of the denominator is 8, which occurs when $x=0$ . As $x$ increases or decreases, the denominator increases without bound.

Finding the Range

To find the range, we need to consider the behavior of the function as $x$ varies. Since the denominator is always positive (or zero), the function is always defined for all real values of $x$ . However, the numerator is a constant, and the function is a rational function, so we need to consider the sign of the function.

Sign of the Function

The function is $y=\frac{-24}{x^2+8}$ . Since the numerator is negative and the denominator is always positive (or zero), the function is always negative. Therefore, the range of the function is all real numbers less than 0.

Conclusion

In conclusion, the range of the function $y=\frac{-24}{x^2+8}$ is all real numbers less than 0. This is because the function is always defined for all real values of $x$ , and the function is always negative due to the sign of the numerator and the denominator.

Answer

The correct answer is A. All real numbers less than 0.

Discussion

The range of a function is an important concept in mathematics, and it has many applications in various fields, such as physics, engineering, and economics. Understanding the range of a function can help us make predictions and model real-world phenomena.

Common Mistakes

When finding the range of a function, it's common to make mistakes. Here are some common mistakes to avoid:

Not considering the sign of the function: Make sure to consider the sign of the function, as it can affect the range.
Not considering the behavior of the denominator: Make sure to consider the behavior of the denominator, as it can affect the range.
Not considering the behavior of the numerator: Make sure to consider the behavior of the numerator, as it can affect the range.

Tips and Tricks

Here are some tips and tricks to help you find the range of a function:

Use algebraic manipulations: Use algebraic manipulations to simplify the function and make it easier to analyze.
Use graphical methods: Use graphical methods, such as plotting the function, to visualize the behavior of the function.
Use numerical methods: Use numerical methods, such as numerical integration, to approximate the range of the function.

Conclusion

Q: What is the range of a function?

A: The range of a function is the set of all possible output values it can produce for the given input values.

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

A: To find the range of a function, you need to consider the behavior of the function as the input value varies. This includes considering the sign of the function, the behavior of the denominator, and the behavior of the numerator.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can the range of a function be empty?

A: Yes, the range of a function can be empty. This occurs when the function is a constant function that is equal to zero.

Q: Can the range of a function be infinite?

A: Yes, the range of a function can be infinite. This occurs when the function is a rational function with a non-constant numerator and a non-constant denominator.

Q: How do I determine if the range of a function is bounded or unbounded?

A: To determine if the range of a function is bounded or unbounded, you need to consider the behavior of the function as the input value varies. If the function approaches a finite limit as the input value approaches infinity, then the range is bounded. If the function approaches infinity as the input value approaches infinity, then the range is unbounded.

Q: Can the range of a function be a single value?

A: Yes, the range of a function can be a single value. This occurs when the function is a constant function.

Q: Can the range of a function be a set of values?

A: Yes, the range of a function can be a set of values. This occurs when the function is a rational function with a non-constant numerator and a non-constant denominator.

Q: How do I graph the range of a function?

A: To graph the range of a function, you need to plot the function on a coordinate plane and identify the set of all possible output values.

Q: Can the range of a function be a function of another variable?

A: Yes, the range of a function can be a function of another variable. This occurs when the function is a composite function.

Q: Can the range of a function be a vector?

A: Yes, the range of a function can be a vector. This occurs when the function is a vector-valued function.

Q: How do I find the range of a function with a square root?

A: To find the range of a function with a square root, you need to consider the behavior of the function as the input value varies. You also need to consider the domain of the function, as the square root function is only defined for non-negative values.

Q: Can the range of a function be a complex number?

A: Yes, the range of a function can be a complex number. This occurs when the function is a complex-valued function.

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

A: To find the range of a function with a logarithm, you need to consider the behavior of the function as the input value varies. You also need to consider the domain of the function, as the logarithm function is only defined for positive values.

Q: Can the range of a function be a function of multiple variables?

A: Yes, the range of a function can be a function of multiple variables. This occurs when the function is a multivariable function.

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

A: To find the range of a function with a trigonometric function, you need to consider the behavior of the function as the input value varies. You also need to consider the domain of the function, as the trigonometric functions are only defined for certain values.

Conclusion

In conclusion, the range of a function is an important concept in mathematics, and it has many applications in various fields. By understanding the range of a function, we can make predictions and model real-world phenomena. Remember to consider the sign of the function, the behavior of the denominator, and the behavior of the numerator when finding the range of a function.