Let's graph the function [tex]\( g(x) = -3 + \log_3(x) \)[/tex].
### Step-by-Step Solution
1. Identify the Function and its Components:
- The function provided is [tex]\( g(x) = -3 + \log_3(x) \)[/tex].
2. Generate [tex]\( x \)[/tex]-values:
- We will generate [tex]\( x \)[/tex]-values over an interval. For this example, let's consider values ranging from 0.1 to 10.
3. Calculate Corresponding [tex]\( g(x) \)[/tex]-values:
- We need to calculate the corresponding [tex]\( g(x) \)[/tex]-values for each of these [tex]\( x \)[/tex]-values using the function's formula [tex]\( g(x) = -3 + \log_3(x) \)[/tex].
- Here is a subset of calculated values:
- For [tex]\( x = 0.1 \)[/tex], [tex]\( g(x) \approx -5.096 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( g(x) = -3 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( g(x) = -2 \)[/tex]
- For [tex]\( x = 10 \)[/tex], [tex]\( g(x) \approx 0.905 \)[/tex]
4. Choose Two Points for Plotting:
- We will use two points from the computed values:
- Point 1: [tex]\( (1, -3) \)[/tex]
- Point 2: [tex]\( (3, -2) \)[/tex]
5. Draw the Asymptote:
- The vertical asymptote of the logarithmic function [tex]\( g(x) = \log_3(x) \)[/tex] is at [tex]\( x = 0 \)[/tex]. This asymptote will be the line [tex]\( x = 0 \)[/tex].
6. Plot the Points and Asymptote:
- Points:
- Plot [tex]\( (1, -3) \)[/tex]
- Plot [tex]\( (3, -2) \)[/tex]
- Asymptote:
- Draw a vertical line at [tex]\( x = 0 \)[/tex].
7. Graph the Function:
- Using the [tex]\( x \)[/tex]-values and the corresponding [tex]\( g(x) \)[/tex]-values, plot the curve through these points.
Here's a simplified representation of the graph with points and asymptote:
### Graph Representation:
- Vertical Asymptote: [tex]\( x = 0 \)[/tex]
- Points on the Graph:
- [tex]\( (1, -3) \)[/tex]
- [tex]\( (3, -2) \)[/tex]
```
y
|
|
| .
|
| (3, -2)
| .
|
| .
|
|_______(1, -3)_________________________ x
|
```
Note:
- The plot above is a simplified sketch. In practice, you would use graphing software or tools to plot a more precise curve.
This plot visually represents [tex]\( g(x) = -3 + \log_3(x) \)[/tex] with the chosen points and the asymptote.
