To determine the domain and range of the function [tex]\( f(x) = -\log(5-x) + 9 \)[/tex], let’s analyze it step by step.
### 1. Domain
The domain of the function is determined by the values of [tex]\( x \)[/tex] for which the function is defined.
The logarithmic function [tex]\(\log(5-x)\)[/tex] is defined when its argument [tex]\( (5-x) \)[/tex] is positive, i.e., [tex]\( 5 - x > 0 \)[/tex].
Solving this inequality:
[tex]\[
5 - x > 0 \\
5 > x \\
x < 5
\][/tex]
So, the domain of [tex]\( f(x) \)[/tex] is [tex]\( x < 5 \)[/tex].
### 2. Range
To find the range, we need to consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches the boundary of its domain and within its domain.
#### Behavior as [tex]\( x \)[/tex] approaches 5 from the left:
As [tex]\( x \)[/tex] approaches 5 from the left ([tex]\( x \to 5^- \)[/tex]), [tex]\( 5 - x \)[/tex] approaches 0 from the positive side. Since [tex]\(\log(0^+) \to -\infty\)[/tex],
[tex]\[
\log(5 - x) \to -\infty
\][/tex]
Hence,
[tex]\[
-\log(5 - x) \to \infty
\][/tex]
Adding 9 to this,
[tex]\[
f(x) \to \infty
\][/tex]
#### Behavior within the domain ([tex]\( x < 5 \)[/tex]):
As [tex]\( x \)[/tex] decreases, for example, when [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex] within the allowed domain which is [tex]\( x < 5 \)[/tex], the value of [tex]\(\log(5 - x)\)[/tex] becomes very large positively since [tex]\( 5 - x \)[/tex] becomes large. Thus:
[tex]\[
\log(5 - x) \to \infty
\][/tex]
[tex]\[
-\log(5 - x) \to -\infty
\][/tex]
Adding 9 to this,
[tex]\[
f(x) \to -\infty
\][/tex]
Putting these observations together, we see that as [tex]\(x\)[/tex] varies over its domain ([tex]\(-\infty < x < 5\)[/tex]), the function [tex]\( f(x) \)[/tex] covers all values from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex].
Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
### Conclusion
The solution to the question is:
- Domain: [tex]\( x < 5 \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
Thus, the correct answer is:
[tex]\( \text{domain: } x < 5 \text{ range: } (-\infty, \infty) \)[/tex].
