Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Certainly! Let's analyze and graph the function [tex]\( f(x) = \log(2 - x) \)[/tex]. Here are the steps to graph the function and determine its domain and asymptote:
1. Understand the Function:
[tex]\( f(x) = \log(2 - x) \)[/tex] is a logarithmic function, specifically the natural logarithm.
2. Domain of the Function:
The logarithmic function [tex]\( \log(y) \)[/tex] is defined only for [tex]\( y > 0 \)[/tex]. Hence, for [tex]\( f(x) = \log(2 - x) \)[/tex], we need [tex]\( 2 - x > 0 \)[/tex].
[tex]\[ 2 - x > 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x < 2 \][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 2) \)[/tex].
3. Vertical Asymptote:
The function [tex]\( f(x) = \log(2 - x) \)[/tex] approaches negative infinity as [tex]\( x \)[/tex] approaches 2 from the left because [tex]\(\log(0)\)[/tex] is undefined and [tex]\(\log\)[/tex] of a number close to 0 is very large in negative value.
Hence, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
4. Graph the Function:
- Draw the vertical asymptote [tex]\( x = 2 \)[/tex].
- Plot some points within the domain [tex]\( (-\infty, 2) \)[/tex] to understand the behavior of the function. For example:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \log(2 - 1) = \log(1) = 0 \][/tex]
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log(2 - 0) = \log(2) \approx 0.693 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \log(2 + 1) = \log(3) \approx 1.098 \][/tex]
- As [tex]\( x \)[/tex] approaches 2 from the left, [tex]\( f(x) \)[/tex] decreases sharply.
Now, plot these points on the graph and draw a curve that approaches [tex]\( x = 2 \)[/tex] as a vertical asymptote:
- The graph approaches the vertical line [tex]\( x = 2 \)[/tex] and never crosses it.
- The curve passes through the point (1, 0).
- As [tex]\( x \)[/tex] decreases, the value of [tex]\( f(x) \)[/tex] continues to increase, but very slowly because of the logarithmic nature of the function.
Here's a rough graph representation:
```
y
^
|
|
|
| *
|
---------------------------------------------------> x
0 |
|
|
|
|
|
```
As we can see from the graph, the function [tex]\( f(x) \)[/tex] rises slowly as [tex]\( x \)[/tex] decreases (towards negative infinity) and approaches the asymptote [tex]\( x = 2 \)[/tex] as [tex]\( x \)[/tex] increases towards 2 from the left.
To summarize:
- Domain: [tex]\( (-\infty, 2) \)[/tex]
- Equation of the Vertical Asymptote: [tex]\( x = 2 \)[/tex]
1. Understand the Function:
[tex]\( f(x) = \log(2 - x) \)[/tex] is a logarithmic function, specifically the natural logarithm.
2. Domain of the Function:
The logarithmic function [tex]\( \log(y) \)[/tex] is defined only for [tex]\( y > 0 \)[/tex]. Hence, for [tex]\( f(x) = \log(2 - x) \)[/tex], we need [tex]\( 2 - x > 0 \)[/tex].
[tex]\[ 2 - x > 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x < 2 \][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 2) \)[/tex].
3. Vertical Asymptote:
The function [tex]\( f(x) = \log(2 - x) \)[/tex] approaches negative infinity as [tex]\( x \)[/tex] approaches 2 from the left because [tex]\(\log(0)\)[/tex] is undefined and [tex]\(\log\)[/tex] of a number close to 0 is very large in negative value.
Hence, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
4. Graph the Function:
- Draw the vertical asymptote [tex]\( x = 2 \)[/tex].
- Plot some points within the domain [tex]\( (-\infty, 2) \)[/tex] to understand the behavior of the function. For example:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \log(2 - 1) = \log(1) = 0 \][/tex]
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \log(2 - 0) = \log(2) \approx 0.693 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \log(2 + 1) = \log(3) \approx 1.098 \][/tex]
- As [tex]\( x \)[/tex] approaches 2 from the left, [tex]\( f(x) \)[/tex] decreases sharply.
Now, plot these points on the graph and draw a curve that approaches [tex]\( x = 2 \)[/tex] as a vertical asymptote:
- The graph approaches the vertical line [tex]\( x = 2 \)[/tex] and never crosses it.
- The curve passes through the point (1, 0).
- As [tex]\( x \)[/tex] decreases, the value of [tex]\( f(x) \)[/tex] continues to increase, but very slowly because of the logarithmic nature of the function.
Here's a rough graph representation:
```
y
^
|
|
|
| *
|
---------------------------------------------------> x
0 |
|
|
|
|
|
```
As we can see from the graph, the function [tex]\( f(x) \)[/tex] rises slowly as [tex]\( x \)[/tex] decreases (towards negative infinity) and approaches the asymptote [tex]\( x = 2 \)[/tex] as [tex]\( x \)[/tex] increases towards 2 from the left.
To summarize:
- Domain: [tex]\( (-\infty, 2) \)[/tex]
- Equation of the Vertical Asymptote: [tex]\( x = 2 \)[/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
