To graph the logarithmic function [tex]\( g(x) = \log_4(x - 1) \)[/tex], let's follow the steps systematically.
### Step 1: Understanding the Function
The function [tex]\( g(x) = \log_4(x - 1) \)[/tex] is a logarithmic function with base 4. The input (argument) to the logarithm is [tex]\( x - 1 \)[/tex].
### Step 2: Identifying the Domain and Range
For the logarithmic function [tex]\( \log_4(x - 1) \)[/tex] to be defined, the argument [tex]\( x - 1 \)[/tex] must be greater than 0.
So, we need:
[tex]\[ x - 1 > 0 \][/tex]
[tex]\[ x > 1 \][/tex]
Thus, the domain of [tex]\( g(x) = \log_4(x - 1) \)[/tex] is:
[tex]\[ (1, \infty) \][/tex]
The range of a logarithmic function in the form of [tex]\( \log_b(f(x)) \)[/tex] for [tex]\( b > 1 \)[/tex] is all real numbers since logarithmic functions can take any real value. Therefore, the range is:
[tex]\[ (-\infty, \infty) \][/tex]
### Step 3: Finding Points to Plot
To plot points on the graph, we choose values of [tex]\( x \)[/tex] such that the function is easy to calculate:
1. Choose [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = \log_4(2 - 1) = \log_4(1) = 0
\][/tex]
The point is [tex]\((2, 0)\)[/tex].
2. Choose [tex]\( x = 5 \)[/tex]:
[tex]\[
g(5) = \log_4(5 - 1) = \log_4(4) = 1
\][/tex]
The point is [tex]\((5, 1)\)[/tex].
### Step 4: Identifying the Asymptote
Since the function [tex]\( g(x) = \log_4(x - 1) \)[/tex] involves [tex]\( x - 1 \)[/tex], the logarithm becomes undefined when [tex]\( x = 1 \)[/tex]. Therefore, there is a vertical asymptote at:
[tex]\[ x = 1 \][/tex]
### Step 5: Graphing the Function
1. Plot the points [tex]\((2, 0)\)[/tex] and [tex]\((5, 1)\)[/tex] on the coordinate plane.
2. Draw the vertical asymptote [tex]\( x = 1 \)[/tex] as a dashed line.
3. Sketch the curve of [tex]\( g(x) = \log_4(x - 1) \)[/tex], which passes through the points you plotted and approaches the vertical asymptote [tex]\( x = 1 \)[/tex] from the right.
### Graph Description:
- The curve starts from just above the x-axis when [tex]\( x \)[/tex] is very close to 1 (but greater than 1).
- It passes through the point [tex]\((2, 0)\)[/tex].
- It increases slowly and passes through the point [tex]\((5, 1)\)[/tex].
- The curve continues to rise, getting more gradual as [tex]\( x \)[/tex] increases.
### Domain and Range in Interval Notation:
- Domain: [tex]\((1, \infty)\)[/tex]
- Range: [tex]\((-\infty, \infty)\)[/tex]
By following these steps, you can successfully graph the logarithmic function [tex]\( g(x) = \log_4(x - 1) \)[/tex], plot the points [tex]\((2, 0)\)[/tex] and [tex]\((5, 1)\)[/tex], and draw the vertical asymptote at [tex]\( x = 1 \)[/tex].
