Question

what are the domain and range of $f(x) = \log(x + 6) - 4$?\
\
\\(\circ\\) domain: $x > -6$; range: $y > 4$\
\\(\circ\\) domain: $x > -6$; range: all real numbers\
\\(\circ\\) domain: $x > 6$; range: $y > -4$\
\\(\circ\\) domain: $x > 6$; range: all real numbers

Explanation:

Step1: Find the domain of the logarithmic function

For a logarithmic function \( f(x)=\log(a) \), the argument \( a \) must be greater than 0. In the function \( f(x)=\log(x + 6)-4 \), the argument is \( x + 6 \). So we set up the inequality:
\( x+6>0 \)
Solving for \( x \), we subtract 6 from both sides:
\( x>-6 \)
So the domain is \( x > - 6 \).

Step2: Find the range of the logarithmic function

The parent function of a logarithmic function \( y = \log(x) \) has a range of all real numbers. When we perform transformations on the logarithmic function, such as vertical shifts (in this case, subtracting 4, which is a vertical shift down by 4 units), the range of the function still remains all real numbers. Because a vertical shift does not restrict the set of possible \( y \)-values (outputs) of the function. The function \( f(x)=\log(x + 6)-4 \) is just a horizontal shift (left by 6 units) and a vertical shift (down by 4 units) of the parent logarithmic function, and these transformations do not change the fact that the range is all real numbers.

Answer:

B. domain: \( x > -6 \); range: all real numbers