Question

select the correct answer. what is the domain and range of this function? r(x) = \log_{5}(x - 2) + 1 a. domain: x > -2, range: all positive real numbers b. domain: x > 2, range: all positive real numbers c. domain: x > -2, range: all real numbers d. domain: x > 2, range: all real numbers

Explanation:

Step1: Find the domain of the logarithmic function

For a logarithmic function \( \log_b(u) \), the argument \( u \) must be greater than 0. In the function \( r(x)=\log_5(x - 2)+1 \), the argument is \( x - 2 \). So we set up the inequality:
\( x - 2>0 \)
Solving for \( x \), we add 2 to both sides of the inequality:
\( x>2 \)
So the domain of the function is \( x > 2 \).

Step2: Find the range of the logarithmic function

The range of a logarithmic function \( \log_b(u) \) (where \( b>0, b
eq1 \)) is all real numbers, regardless of the vertical shift or horizontal shift. The function \( r(x)=\log_5(x - 2)+1 \) is a logarithmic function with a vertical shift of 1, but the vertical shift does not affect the range (it just shifts the graph up or down). The range of \( \log_5(x - 2) \) is all real numbers, and adding 1 to it still keeps the range as all real numbers.

Answer:

D. domain: \( x > 2 \); range: all real numbers