- Identify that the argument \(x-1\) must be greater than 0, resulting in a domain of \(x > 1\).
- Recognize that the range of the logarithmic function \(\log(u)\) is all real numbers, and this remains unchanged by a vertical shift.
- Conclude that the correct option is \(\boxed{b}\).
Explanation
- Determining the Domain
We are given the function \(f(x) = \log(x - 1) + 2\). A logarithmic function \(\log(u)\) is only defined when the argument \(u\) is strictly greater than 0. In this case, the argument of the logarithm is \(u = x - 1\). Therefore, for the function to be defined, we must have \(x - 1 > 0\), which simplifies to \(x > 1\). This condition defines the domain of the function.
- Determining the Range
Next, we analyze the range of the function. The basic logarithmic function \(y = \log(u)\) takes all real numbers as its output (range) as \(u\) varies from 0 to positive infinity. Adding a constant, in this case 2, vertically shifts the entire graph of the function but does not change the set of possible output values. Since the output of \(\log(u)\) is all real numbers, the range of \(f(x) = \log(x - 1) + 2\) remains the set of all real numbers.
- Matching with Provided Options
Comparing our results with the given options:
Adomain: all real numbers; range: all real numbers
Bdomain: \(x > 1\); range: all real numbers
Cdomain: all real numbers; range: \(y > 1\)
Ddomain: \(x > 1\); range: \(y > 2\)
Our findings indicate that the domain is \(x > 1\) and the range is all real numbers, which corresponds to option b.
Examples
The concept of the domain of a logarithmic function is essential in fields like finance and science. For example, when calculating the time it takes for an investment to grow to a certain amount using the formula \(t = \log(A/P) / \log(1+r)\), the expression inside the logarithm must be positive. Understanding the domain of these functions ensures that our calculations are valid and meaningful in real-world contexts, helping us avoid undefined mathematical results.