To determine the domain and range of the function [tex]\( f(x) = \log(x - 1) + 2 \)[/tex], we need to carefully analyze the behavior of the logarithmic function [tex]\( \log(x - 1) \)[/tex] and how it is affected by the addition of 2.
Step-by-Step Analysis:
1. Domain:
- The argument of the logarithm [tex]\( x - 1 \)[/tex] must be greater than 0 for the logarithm to be defined.
- Therefore, [tex]\( x - 1 > 0 \implies x > 1 \)[/tex].
- This means the domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 1 \)[/tex].
2. Range:
- The logarithmic function [tex]\( \log(x - 1) \)[/tex] can take any real number value.
- So, [tex]\( \log(x - 1) \)[/tex] can be any real number from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
- When we add 2 to [tex]\( \log(x - 1) \)[/tex], we are shifting the entire set of values of [tex]\( \log(x - 1) \)[/tex] up by 2 units.
- Thus, the range of [tex]\( f(x) = \log(x - 1) + 2 \)[/tex] will be all real numbers greater than 2.
Conclusion:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 1 \)[/tex].
- The range of [tex]\( f(x) \)[/tex] is [tex]\( y > 2 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\text{Domain: } x > 1; \text{ Range: } y > 2
\][/tex]
