Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.