Certainly! Let's graph the logarithmic function [tex]\( g(x) = 2 + \log_4(x - 1) \)[/tex]. We will find and plot two points on the graph, and describe its domain and range. We will also identify any asymptotes.
### Step-by-Step Solution
1. Identify the Domain:
The argument of the logarithm must be positive:
[tex]\[
x - 1 > 0 \implies x > 1
\][/tex]
So, the domain of [tex]\( g(x) \)[/tex] is [tex]\( (1, \infty) \)[/tex].
2. Calculate Specific Points:
Let's choose two values of [tex]\( x \)[/tex] that satisfy the domain [tex]\( x > 1 \)[/tex].
- Point 1: [tex]\( x = 2 \)[/tex]
[tex]\[
g(2) = 2 + \log_4(2 - 1) = 2 + \log_4(1) = 2 + 0 = 2
\][/tex]
So, one point is [tex]\( (2, 2) \)[/tex].
- Point 2: [tex]\( x = 5 \)[/tex]
[tex]\[
g(5) = 2 + \log_4(5 - 1) = 2 + \log_4(4) = 2 + 1 = 3
\][/tex]
So, the second point is [tex]\( (5, 3) \)[/tex].
3. Vertical Asymptote:
The logarithmic function [tex]\( \log_4(x - 1) \)[/tex] has a vertical asymptote where the argument is zero:
[tex]\[
x - 1 = 0 \implies x = 1
\][/tex]
Thus, there is a vertical asymptote at [tex]\( x = 1 \)[/tex].
4. Range of the Function:
Logarithmic functions can take any real value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex] as [tex]\( x \)[/tex] increases beyond the domain restriction. Since we are adding 2 to the logarithmic function, the overall range is still all real numbers:
[tex]\[
g(x) \in (-\infty, \infty)
\][/tex]
### Summary of Information:
- Domain: [tex]\( (1, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
- Vertical Asymptote: [tex]\( x = 1 \)[/tex]
- Points to Plot: [tex]\( (2, 2) \)[/tex] and [tex]\( (5, 3) \)[/tex]
### Graph:
Here is how you can visualize the function:
1. Draw the x-axis and y-axis.
2. Draw a dashed vertical line at [tex]\( x = 1 \)[/tex] to represent the vertical asymptote.
3. Plot the points [tex]\( (2, 2) \)[/tex] and [tex]\( (5, 3) \)[/tex].
4. Sketch the curve that passes through these points, showing the typical shape of a logarithmic function approaching the vertical asymptote from the right.
This will help you visualize the overall behavior of the function [tex]\( g(x) = 2 + \log_4(x - 1) \)[/tex].
