Restrictions Imposed by the Rational Function

Oct 13, 2025 by miodeya.com 48 views

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll explore the domain of the function $$y=\ln \left(\frac{-x+3}{2}\right)$$ and determine the correct answer from the given options.

What is the Natural Logarithm Function?

The natural logarithm function, denoted by ln(x), is a mathematical function that takes a positive real number as input and returns the logarithm of that number to the base e (approximately 2.718). The natural logarithm function is defined only for positive real numbers, and its domain is the set of all positive real numbers.

The Given Function: $$y=\ln \left(\frac{-x+3}{2}\right)$$

The given function is a composition of the natural logarithm function and a rational function. The rational function is defined as $$\frac{-x+3}{2}$$, and the natural logarithm function is applied to this rational function. To determine the domain of the given function, we need to consider the restrictions imposed by both the rational function and the natural logarithm function.

Restrictions Imposed by the Rational Function

The rational function $$\frac{-x+3}{2}$$ is defined for all real numbers except when the denominator is zero. The denominator is zero when $$-x+3=0$$, which occurs when $$x=3$$. Therefore, the rational function is undefined at $$x=3$$.

Restrictions Imposed by the Natural Logarithm Function

The natural logarithm function is defined only for positive real numbers. Therefore, the input to the natural logarithm function, which is the rational function $$\frac{-x+3}{2}$$, must be positive. This means that $$\frac{-x+3}{2}>0$$.

Solving the Inequality: $$\frac{-x+3}{2}>0$$

To solve the inequality $$\frac{-x+3}{2}>0$$, we can multiply both sides by 2 to eliminate the fraction. This gives us $$-x+3>0$$. We can then add x to both sides to get $$3>x$$. Finally, we can subtract 3 from both sides to get $$x<3$$.

Conclusion

In conclusion, the domain of the function $$y=\ln \left(\frac{-x+3}{2}\right)$$ is the set of all real numbers x such that $$x<3$$. This means that the correct answer is:

A. $$x\ \textless \ 2$$

However, this is not the correct answer. The correct answer is actually:

C. $$x\ \textless \ 3$$

This is because the domain of the function is the set of all real numbers x such that $$x<3$$, not $$x<2$$. Therefore, the correct answer is C.

Final Answer

Introduction

In our previous article, we explored the concept of the domain of a function and determined the domain of the function $$y=\ln \left(\frac{-x+3}{2}\right)$$. In this article, we'll answer some frequently asked questions about the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the restrictions imposed by the function itself. For example, if the function involves a square root, you need to consider the values of x that make the expression under the square root non-negative. Similarly, if the function involves a logarithm, you need to consider the values of x that make the expression inside the logarithm positive.

Q: What are some common restrictions that affect the domain of a function?

A: Some common restrictions that affect the domain of a function include:

Division by zero: If a function involves division by zero, it is undefined at that point.
Square root of a negative number: If a function involves the square root of a negative number, it is undefined at that point.
Logarithm of a non-positive number: If a function involves the logarithm of a non-positive number, it is undefined at that point.
Rational expressions: If a function involves a rational expression, it is undefined at points where the denominator is zero.

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

A: To determine the domain of a rational function, you need to consider the values of x that make the denominator zero. These values are called the zeros of the denominator. The domain of the rational function is the set of all real numbers x except the zeros of the denominator.

Q: How do I determine the domain of a function with a square root?

A: To determine the domain of a function with a square root, you need to consider the values of x that make the expression under the square root non-negative. These values are called the domain of the square root function.

Q: How do I determine the domain of a function with a logarithm?

A: To determine the domain of a function with a logarithm, you need to consider the values of x that make the expression inside the logarithm positive. These values are called the domain of the logarithmic function.

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 (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

Conclusion

In conclusion, determining the domain of a function is an essential step in understanding the behavior of the function. By considering the restrictions imposed by the function itself, you can determine the domain of the function and ensure that you are working with a well-defined function.

Final Answer

The final answer is that the domain of a function is the set of all possible input values (x-values) for which the function is defined.