Let's analyze the function [tex]\( f(x) = -\log (5-x) + 9 \)[/tex] step by step to determine its domain and range.
Step 1: Determine the domain
The function [tex]\( f(x) = -\log(5-x) + 9 \)[/tex] involves a logarithm [tex]\(\log(5-x)\)[/tex]. The logarithmic function is defined only for positive arguments. Therefore, the argument of the logarithm must be positive:
[tex]\[ 5 - x > 0 \][/tex]
Solving the inequality:
[tex]\[ x < 5 \][/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ x < 5 \][/tex]
Step 2: Determine the range
To find the range of [tex]\( f(x) \)[/tex], consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] varies within the domain [tex]\( x < 5 \)[/tex].
Rewrite the function:
[tex]\[ f(x) = -\log(5-x) + 9 \][/tex]
As [tex]\( x \)[/tex] approaches 5 from the left:
[tex]\[ 5 - x \rightarrow 0^+ \][/tex]
[tex]\[ \log(5 - x) \rightarrow \log(0^+) = -\infty \][/tex]
Therefore,
[tex]\[ -\log(5 - x) \rightarrow \infty \][/tex]
Thus,
[tex]\[ f(x) \rightarrow \infty \][/tex]
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 5 - x \rightarrow \infty \][/tex]
[tex]\[ \log(5 - x) \rightarrow \log(\infty) = \infty \][/tex]
Therefore,
[tex]\[ -\log(5 - x) \rightarrow -\infty \][/tex]
Thus,
[tex]\[ f(x) = -\log(5 - x) + 9 \rightarrow -\infty + 9 = 9 \][/tex]
However, for the smallest value within the domain, let's evaluate at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\log(5 - 0) + 9 \][/tex]
[tex]\[ f(0) = -\log(5) + 9 \approx 7.39 \][/tex]
This indicates the smallest value of [tex]\( f(x) \)[/tex]. Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ [7.39, \infty) \][/tex]
Concluding:
Domain: [tex]\( x < 5 \)[/tex]
Range: [tex]\( y \geq 7.39 \)[/tex]
