Let's determine the domain and range of the function [tex]\( f(x) = \log(x-4) - 3 \)[/tex].
### Finding the Domain:
The function involves a logarithmic component [tex]\(\log(x-4)\)[/tex]. For the logarithm function to be defined, its argument must be positive. Therefore, we need:
[tex]\[ x - 4 > 0 \][/tex]
[tex]\[ x > 4 \][/tex]
This inequality tells us that [tex]\( x \)[/tex] must be greater than 4. Hence, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x > 4 \)[/tex].
In interval notation, the domain is:
[tex]\[ (4, \infty) \][/tex]
### Finding the Range:
Next, let’s determine the range of the function. The expression [tex]\(\log(x-4)\)[/tex] is a logarithmic function, which can take any real value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
Since the function [tex]\( f(x) \)[/tex] is defined as [tex]\(\log(x-4) - 3\)[/tex], we are shifting the logarithmic function vertically downward by 3 units. Mathematically, this does not alter the range of the logarithmic function itself, as a vertical shift only translates the values but does not constrain the output:
- The original range of [tex]\(\log(x-4)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
- After shifting by [tex]\(-3\)[/tex], the range of [tex]\( \log(x-4) - 3 \)[/tex] also remains [tex]\((-\infty, \infty)\)[/tex].
In conclusion, the range of [tex]\( f(x) = \log(x-4) - 3 \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Summary:
- Domain: [tex]\( (4, \infty) \)[/tex]
- Range: [tex]\((- \infty, \infty)\)[/tex]
Thus, the domain and range of the function [tex]\( f(x) = \log(x-4) - 3 \)[/tex] are:
[tex]\[
\text{Domain: } (4, \infty)
\][/tex]
[tex]\[
\text{Range: } (- \infty, \infty)
\][/tex]
