- Determine the domain by solving the condition \(x-1 > 0\), which gives \(x > 1\).
- Recognize that the logarithmic function \(\log(u)\) maps \((0, \infty)\) to \((-\infty, \infty)\).
- Conclude that the range is all real numbers, since adding a constant does not change the set of output values.
- Match these results to option \(\boxed{b}\).
Explanation
- Determining the Domain
To determine the domain of the function \(f(x) = \log(x-1) + 2\), we must consider the definition of the logarithmic function. The function \(\log(u)\) is defined if and only if the argument \(u\) is strictly positive, meaning \(u > 0\). In our function, the argument is \(x-1\). Therefore, we set up the inequality \(x-1 > 0\), which simplifies to \(x > 1\). This means the domain of the function is all real numbers \(x\) greater than 1.
- Determining the Range
To determine the range of the function, we examine the behavior of the expression \(\log(u)\) as \(u\) varies over its domain \((0, \infty)\). The function \(\log(u)\) is continuous and maps the interval \((0, \infty)\) to \((-\infty, \infty)\). When we shift this function by adding a constant, such as the \(+ 2\) in \(f(x) = \log(x-1) + 2\), the set of all possible output values remains the entire set of real numbers. Thus, the range of \(f(x)\) is all real numbers.
- Matching with Options and Conclusion
Comparing our findings to the given options:
ADomain: all real numbers; Range: all real numbers.
BDomain: \(x > 1\); Range: all real numbers.
CDomain: \(x > 1\); Range: \(y > 2\).
Our analysis shows that the domain is \(x > 1\) and the range is all real numbers, which matches option (b). The final answer is \(\boxed{b}\).
Examples
Logarithmic functions are used in many real-world applications where values vary across several orders of magnitude. For example, the pH scale, which measures the acidity or alkalinity of a substance, is defined as \(pH = -\log_{10}[H^+]\), where \([H^+]\) is the concentration of hydrogen ions. Just like in our example, this function is only defined for a positive concentration of ions, and it can take on a wide range of values, representing everything from highly acidic battery acid to highly alkaline drain cleaner. Understanding the domain and range of such functions helps scientists determine the limits of measurement and the scope of the phenomena they are studying.