To find the domain and range of the function [tex]\( f(x) = \log(x-4) - 3 \)[/tex], let's break down the problem step-by-step.
### Finding the Domain:
1. Identify the argument of the logarithmic function: The logarithmic function [tex]\(\log(x-4)\)[/tex] inside [tex]\( f(x) \)[/tex] must have a positive argument.
2. Set up the inequality for the argument: Since the logarithm is only defined for positive values, we need:
[tex]\[
x - 4 > 0
\][/tex]
3. Solve the inequality:
[tex]\[
x > 4
\][/tex]
Therefore, the smallest value that [tex]\(x\)[/tex] can take is just greater than 4. There is no upper limit to the value of [tex]\(x\)[/tex].
4. Express the domain in interval notation:
[tex]\[
(4, \infty)
\][/tex]
Thus, the domain of [tex]\( f(x) = \log(x-4) - 3 \)[/tex] is [tex]\((4, \infty)\)[/tex].
### Finding the Range:
1. Understand the behavior of the logarithmic function: The logarithmic function [tex]\(\log(x-4)\)[/tex] can produce any real number as output; its range is [tex]\((-∞, ∞)\)[/tex].
2. Transform the range by the vertical shift: Subtracting 3 from the logarithm shifts its range down by 3 units, but this shift does not set any specific limits on the transformed values.
3. Determine the new range: Therefore, even after subtracting 3, the output can still be any real number.
Hence, the range of [tex]\( f(x) = \log(x-4) - 3 \)[/tex] is [tex]\((-∞, ∞)\)[/tex].
### Conclusion:
- Domain: [tex]\( (4, \infty) \)[/tex]
- Range: [tex]\((-∞, ∞)\)[/tex]
This means the function [tex]\( f(x) = \log(x-4) - 3 \)[/tex] is defined for all [tex]\(x\)[/tex] values greater than 4, and it can take any real number as its output.
