To graph the logarithmic function [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex], let's go through the steps for plotting and analyzing the function.
### Step 1: Find Points on the Graph
To plot the function, we need to find values for [tex]\( g(x) \)[/tex] at specific points. Let's choose two values for [tex]\( x \)[/tex]:
Point 1:
- Let [tex]\( x = 3 \)[/tex].
- Calculate [tex]\( g(3) = -2 + \log_4(3 + 1) = -2 + \log_4(4) = -2 + 1 = -1 \)[/tex].
So, one point is [tex]\( (3, -1) \)[/tex].
Point 2:
- Let [tex]\( x = 7 \)[/tex].
- Calculate [tex]\( g(7) = -2 + \log_4(7 + 1) = -2 + \log_4(8) \)[/tex].
To calculate [tex]\( \log_4(8) \)[/tex], note that [tex]\( 8 = 4^{3/2} \)[/tex] because [tex]\( 8 = 2^3 \)[/tex] and [tex]\( 4 = 2^2 \)[/tex], therefore:
[tex]\[
\log_4(8) = \log_4(4^{3/2}) = \frac{3}{2}
\][/tex]
Thus,
[tex]\[
g(7) = -2 + \frac{3}{2} = -0.5
\][/tex]
So, another point is [tex]\( (7, -0.5) \)[/tex].
### Step 2: Identify the Asymptote
Logarithmic functions have a vertical asymptote where the argument of the logarithm is zero. For [tex]\( \log_4(x + 1) \)[/tex], the vertical asymptote occurs at:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
So, the vertical asymptote is at [tex]\( x = -1 \)[/tex].
### Step 3: Determine the Domain and Range
Domain: The function [tex]\( g(x) \)[/tex] is defined for values of [tex]\( x \)[/tex] such that [tex]\( x + 1 > 0 \)[/tex]. Therefore, the domain is [tex]\( x > -1 \)[/tex].
In interval notation, the domain is:
[tex]\[ (-1, \infty) \][/tex]
Range: Since the logarithmic function can take any real number value, and multiplying or adding constants do not restrict this, the range of [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex] is all real numbers.
In interval notation, the range is:
[tex]\[ (-\infty, \infty) \][/tex]
### Summary
- Two plotted points: [tex]\( (3, -1) \)[/tex] and [tex]\( (7, -0.5) \)[/tex]
- Vertical asymptote: [tex]\( x = -1 \)[/tex]
- Domain: [tex]\( (-1, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
These elements together provide a full understanding of the graph of the function [tex]\( g(x) = -2 + \log_4(x + 1) \)[/tex].
