Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine which function has an asymptote at [tex]\( x = 5 \)[/tex] and an [tex]\( x \)[/tex]-intercept of [tex]\( (6, 0) \)[/tex], let's examine each option step-by-step.
### Option A: [tex]\( f(x) = \log (x-5) \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log (x-5) \)[/tex] is undefined when [tex]\( x-5 \leq 0 \)[/tex], which means at [tex]\( x = 5 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 5 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log(x-5) = 0 \)[/tex], we get:
[tex]\[ \log(x-5) = 0 \implies x-5 = 10^0 \implies x - 5 = 1 \implies x = 6 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (6, 0) \)[/tex].
Both the conditions (asymptote at [tex]\( x = 5 \)[/tex] and [tex]\( x \)[/tex]-intercept at [tex]\( (6, 0) \)[/tex]) are satisfied by this function.
### Option B: [tex]\( f(x) = \log (x+5) \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log (x+5) \)[/tex] is undefined when [tex]\( x+5 \leq 0 \)[/tex], which means at [tex]\( x = -5 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = -5 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log(x+5) = 0 \)[/tex], we get:
[tex]\[ \log(x+5) = 0 \implies x+5 = 10^0 \implies x + 5 = 1 \implies x = -4 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-4, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Option C: [tex]\( f(x) = \log x - 5 \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log x \)[/tex] is undefined when [tex]\( x \leq 0 \)[/tex], which means at [tex]\( x = 0 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 0 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log x - 5 = 0 \)[/tex], we get:
[tex]\[ \log x - 5 = 0 \implies \log x = 5 \implies x = 10^5 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (10^5, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Option D: [tex]\( f(x) = \log x + 5 \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log x \)[/tex] is undefined when [tex]\( x \leq 0 \)[/tex], which means at [tex]\( x = 0 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 0 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log x + 5 = 0 \)[/tex], we get:
[tex]\[ \log x + 5 = 0 \implies \log x = -5 \implies x = 10^{-5} \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (10^{-5}, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Conclusion
Only option [tex]\( A \)[/tex], [tex]\( f(x) = \log (x-5) \)[/tex], has an asymptote at [tex]\( x = 5 \)[/tex] and an [tex]\( x \)[/tex]-intercept at [tex]\( (6, 0) \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{A} \][/tex]
### Option A: [tex]\( f(x) = \log (x-5) \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log (x-5) \)[/tex] is undefined when [tex]\( x-5 \leq 0 \)[/tex], which means at [tex]\( x = 5 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 5 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log(x-5) = 0 \)[/tex], we get:
[tex]\[ \log(x-5) = 0 \implies x-5 = 10^0 \implies x - 5 = 1 \implies x = 6 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (6, 0) \)[/tex].
Both the conditions (asymptote at [tex]\( x = 5 \)[/tex] and [tex]\( x \)[/tex]-intercept at [tex]\( (6, 0) \)[/tex]) are satisfied by this function.
### Option B: [tex]\( f(x) = \log (x+5) \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log (x+5) \)[/tex] is undefined when [tex]\( x+5 \leq 0 \)[/tex], which means at [tex]\( x = -5 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = -5 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log(x+5) = 0 \)[/tex], we get:
[tex]\[ \log(x+5) = 0 \implies x+5 = 10^0 \implies x + 5 = 1 \implies x = -4 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-4, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Option C: [tex]\( f(x) = \log x - 5 \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log x \)[/tex] is undefined when [tex]\( x \leq 0 \)[/tex], which means at [tex]\( x = 0 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 0 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log x - 5 = 0 \)[/tex], we get:
[tex]\[ \log x - 5 = 0 \implies \log x = 5 \implies x = 10^5 \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (10^5, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Option D: [tex]\( f(x) = \log x + 5 \)[/tex]
1. Asymptote Analysis:
- The logarithmic function [tex]\( \log x \)[/tex] is undefined when [tex]\( x \leq 0 \)[/tex], which means at [tex]\( x = 0 \)[/tex].
- Therefore, there is an asymptote at [tex]\( x = 0 \)[/tex].
2. [tex]\( x \)[/tex]-intercept Analysis:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex].
- Setting [tex]\( \log x + 5 = 0 \)[/tex], we get:
[tex]\[ \log x + 5 = 0 \implies \log x = -5 \implies x = 10^{-5} \][/tex]
- Hence, the [tex]\( x \)[/tex]-intercept is at [tex]\( (10^{-5}, 0) \)[/tex], not [tex]\( (6, 0) \)[/tex].
This option does not satisfy both conditions.
### Conclusion
Only option [tex]\( A \)[/tex], [tex]\( f(x) = \log (x-5) \)[/tex], has an asymptote at [tex]\( x = 5 \)[/tex] and an [tex]\( x \)[/tex]-intercept at [tex]\( (6, 0) \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{A} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
