How to Find the Domain and Range of (y log(x-1) - 1)

Understanding the domain and range of a function is crucial in mathematics, especially when working with logarithmic functions. In this article, we will explore how to determine the domain and range of the function (y log(x-1) - 1). We will break down the steps and use mathematical reasoning to understand each component of this function.

Understanding the Function

The given function is (y log(x-1) - 1). To analyze this function, we need to remember a few key points about the logarithmic function (log x) and its domain and range.

Domain of (y log(x-1))

The domain of the logarithmic function (log x) is (x 0). This means that (x) must be positive for the logarithm to be defined. To extend this to the function (y log(x-1)), we need to ensure that the argument of the logarithm, (x-1), is greater than zero. This translates to:

[x - 1 0]

By solving this inequality:

[x 1]

Thus, the domain of (y log(x-1)) is (1 x infty), or mathematically, (x in (1, infty)).

Appending the constant (-1) to the function, we get (y log(x-1) - 1). Since the addition of a constant does not affect the domain, the domain of the function (y log(x-1) - 1) remains the same:

(mathfrak{D}_f (1, infty))

Range of (y log(x-1) - 1)

To determine the range of (y log(x-1) - 1), we need to consider the properties of the logarithmic function. The range of (y log(x-1)) is all real numbers, as the logarithm can take any positive value. By adding or subtracting a constant, the range does not change. Therefore, the range of (y log(x-1) - 1) is also all real numbers:

(mathfrak{R}_f (-infty, infty))

This means that for any value of (x) in the domain, (f(x)) can take any real number as its output.

Graphical Interpretation

The graph of the function (y log(x-1) - 1) visually confirms the domain and range we have determined. The graph will have a vertical asymptote at (x 1), approaching negative infinity as (x) approaches 1 from the right, and extending to positive infinity as (x) increases. The shape of the graph will be a shifted version of the standard logarithmic function (log x).

View the graph of (y log(x-1) - 1)

Conclusion

By understanding the domain and range of the function (y log(x-1) - 1), we can effectively analyze its behavior and properties. The domain of this function is (1 x infty), and the range is all real numbers, which is a key characteristic of logarithmic functions.

Key Takeaways:

(mathfrak{D}_f (1, infty)) (mathfrak{R}_f (-infty, infty)) Logarithmic functions are only defined for positive arguments. The range of (y log(x-1) - 1) is all real numbers, due to the properties of the logarithmic function.

Understanding these concepts is essential for anyone working with logarithmic functions and their applications in various fields such as science, engineering, and economics.