To determine the domain and range of the function [tex]\( f(x) = \log_5(x-2) + 1 \)[/tex], let's break down the problem step by step.
1. Determining the Domain:
- The logarithmic function [tex]\( \log_b(y) \)[/tex] is defined only for positive values of [tex]\( y \)[/tex]. This means that the argument of the logarithm, [tex]\( (x-2) \)[/tex], must be greater than 0.
- Therefore, the inequality we need to solve is:
[tex]\[
x - 2 > 0
\][/tex]
- Solving this inequality, we get:
[tex]\[
x > 2
\][/tex]
- Hence, the domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 2 \)[/tex].
2. Determining the Range:
- The logarithmic function [tex]\( \log_5(y) \)[/tex] can produce any real number (positive, negative, or zero). When we add 1 to a real number, the result is still a real number.
- Therefore, the expression [tex]\( \log_5(x-2) + 1 \)[/tex] can take any real number value.
- Thus, the range of [tex]\( f(x) \)[/tex] is all real numbers.
Based on this information:
- The domain is [tex]\( x > 2 \)[/tex].
- The range is all real numbers.
The correct answer from the options provided is:
D. domain: [tex]\( x > 2 \)[/tex]; range: all real numbers
