Understanding the Domain and Range of the Function f(x) 1/√(x-1)

The function f(x) 1/√(x-1) presents unique challenges in determining its domain and range. Let's delve into the mathematical reasoning and steps required to find these essential properties.

The Function and Its Constraints

The function g(x) 1/√(x-1) is characterized by a square root in the denominator, making it necessary to impose constraints on the domain of x. This is because the square root of a negative number is not defined in the set of real numbers. Moreover, the denominator cannot be zero since division by zero is undefined.

Domain Analysis

The domain of a function is the set of all input values (x) for which the function is defined. For the function f(x) 1/√(x-1):

We start by ensuring the expression inside the square root is non-negative:

x - 1 0 This implies x 1.

We also need to ensure the denominator is not zero, which is already fulfilled since x - 1 0.

Therefore, the domain of the function is all real numbers greater than 1, which can be written as (1, infty), or using interval notation:

x in (1, infty).

Range Analysis

The range of a function is the set of all output values (y) produced by the function. For f(x) 1/√(x-1), we analyze its behavior as x approaches certain values:

As x 1, the expression inside the square root, x - 1, becomes negative, making the function undefined. As x approaches 1 from the right, √(x-1) approaches 0 from the positive side, making the function approach positive infinity. As x approaches infinity, √(x-1) also approaches infinity, making the function approach 0 from the positive side.

Therefore, the range of the function is all positive real numbers, which can be written as (0, infty).

Mathematical Representation

Here's the formal mathematical representation:

Domain: The domain of the function f(x) 1/√(x-1) is all real numbers greater than 1:

D: x in (1, infty).

Range: The range of the function f(x) 1/√(x-1) is all positive real numbers:

R: y in (0, infty).

Conclusion

The domain of the function f(x) 1/√(x-1) is (1, infty), and the range is (0, infty). This analysis showcases the importance of considering both the non-negativity under the square root and the non-zero denominator to determine the domain and range of a function.

Understanding these properties is crucial for further mathematical analysis and graphing of the function. If you have any more questions, feel free to reach out!