Let's analyze the function [tex]\( f(x) = -\log (5 - x) + 9 \)[/tex] to determine its domain and range.
### Domain:
1. Function Definition:
[tex]\(\log(5 - x)\)[/tex] implies a logarithmic function. The argument of the logarithm (inside the [tex]\(\log\)[/tex]) must be positive because the logarithm of a non-positive number is undefined in the realm of real numbers.
2. Set the Argument [tex]\(5-x\)[/tex] > 0:
[tex]\[
5 - x > 0
\][/tex]
3. Solve the Inequality:
[tex]\[
x < 5
\][/tex]
Therefore, the domain of the function [tex]\( f(x) = - \log (5 - x) + 9 \)[/tex] is [tex]\( x < 5 \)[/tex].
### Range:
1. Logarithmic Function Range:
Recall that the logarithmic function can take any real number as its output, [tex]\((-\infty, +\infty)\)[/tex].
2. Effect of Negative Sign:
When applying the negative sign, [tex]\(-\log(5 - x)\)[/tex], the range still remains all real numbers because a negative can multiply any value.
3. Shifting by +9:
Adding 9 to the entire function will shift the range upward by 9. Thus:
- The minimum value [tex]\(-\log(5 - x)\)[/tex] can attain is negative infinity before adding 9.
- When shifted by +9, any negative value plus 9 remains smaller than 9, and all positive values increased by 9.
- The function [tex]\( f(x) = -\log (5 - x) + 9 \)[/tex] therefore covers the range starting from 9 and extending to infinity.
4. Final Range:
Therefore, [tex]\( f(x) \geq 9 \)[/tex]. The range of the function [tex]\( f(x) = -\log (5 - x) + 9 \)[/tex] is [tex]\( y \geq 9 \)[/tex].
### Conclusion:
Based on our derivation, the correct pair for the domain and range is:
- Domain: [tex]\(x < 5\)[/tex]
- Range: [tex]\(y \geq 9\)[/tex]
So, the correct answer is:
[tex]\[ \text{domain: } x < 5, \text{ range: } y \geq 9 \][/tex]
