Question

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

Explanation:

Step1: Find the domain of the logarithmic function

For a logarithmic function \( f(x)=\log_b(u) \), the argument \( u \) 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:
\( x+6>0 \)
Subtract 6 from both sides of the inequality:
\( x>-6 \)
So the domain is \( x > - 6 \).

Step2: Find the range of the logarithmic function

The parent function of a logarithmic function \( y = \log_b(x) \) 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) or horizontal shifts (adding 6, which is a horizontal shift left by 6 units), the range of the logarithmic function does not change. The range of \( f(x)=\log(x + 6)-4 \) is still all real numbers because vertical and horizontal shifts do not restrict the set of all possible \( y \)-values (outputs) of the logarithmic function.

Answer:

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