Question

For the function y = log(x - 2) +1, which of the following statements is true? O A. The domain is (2,5), and the range is [1,-). B. The domain is (2,0), and the range is all real numbers. C. The domain is (-2,0), and the range is all real numbers. D. The domain is all real numbers, and the range is [1,co). SUBMIT

Answer 1

The given function is a logarithmic function:

$y=\log (x-2)+1$

The domain of a logarithmic function can be found using the fact that the argument of the logarithm must be greater than zero.

Notice that the argument here is x-2. Hence, the domain is found by solving for x when x-2>0:

$\begin{gathered} x-2>0 \\ \text{Add 2 to both sides of the inequality:} \\ x-2+2>0+2\Rightarrow x>2 \\ \text{Write the inequality in interval notation:} \\ (2,\infty) \end{gathered}$

Hence, the domain of the function is (2,∞).

Recall that the Range of any Logarithmic Function is all real numbers.

Since the given function is a logarithmic function, it follows that the range of the given function is all real numbers.

Option B is the correct answer.