Answer :
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]
