Answer :
Let's analyze the features of the function \( g(x) = f(x + 4) + 8 \) where \( f(x) = \log_2 x \).
### Step-by-Step Solution
1. Original Function \( f(x) = \log_2 x \):
- Domain: \( (0, \infty) \)
This is because the logarithmic function is only defined for positive values of \( x \).
- Range: \( (-\infty, \infty) \)
Since a logarithmic function can take any real-number value.
- x-intercept: \( (1, 0) \)
This occurs because \( \log_2(1) = 0 \).
- Vertical Asymptote: \( x = 0 \)
As \( x \) approaches zero from the right, \( \log_2 x \) approaches negative infinity.
2. Transformation to \( g(x) = \log_2(x + 4) + 8 \):
Let’s apply the transformations step by step.
- Horizontal Shift (left by 4 units):
\( g(x) = \log_2(x + 4) + 8 \)
This means replacing \( x \) with \( x + 4 \). The entire graph of \( f(x) \) shifts 4 units to the left.
- Vertical Shift (upward by 8 units):
\( g(x) = \log_2(x + 4) + 8 \)
This means the entire graph of \(f(x)\) will shift upward by 8 units.
3. Features of Transformed Function \( g(x) = \log_2(x + 4) + 8 \):
- Domain:
The domain shifts left by 4 units, so the domain of \( g(x) \) is \( (-4, \infty) \).
- Range:
The range of a logarithmic function is all real numbers, and this range does not change with translation. Thus, the range remains \( (-\infty, \infty) \).
- x-intercept:
The x-intercept of \( g(x) \) can be found by setting \( g(x) = 0 \):
[tex]\[ \log_2(x + 4) + 8 = 0 \implies \log_2(x + 4) = -8. \][/tex]
Converting this to exponential form, we get:
[tex]\[ x + 4 = 2^{-8} \implies x + 4 = 0.00390625 \implies x = 0.00390625 - 4 \implies x = -3.99609375. \][/tex]
Thus, the x-intercept is \( (-3.99609375, 0) \).
- Vertical Asymptote:
The vertical asymptote will shift left by 4 units, from \( x = 0 \) to \( x = -4 \).
### Summary of the Features:
- Domain: \( (-4, \infty) \).
- Range: \( (-\infty, \infty) \).
- x-intercept: \( (-3.99609375, 0) \).
- Vertical Asymptote: \( x = -4 \).
These are the key features of the function [tex]\( g(x) = \log_2(x + 4) + 8 \)[/tex].
### Step-by-Step Solution
1. Original Function \( f(x) = \log_2 x \):
- Domain: \( (0, \infty) \)
This is because the logarithmic function is only defined for positive values of \( x \).
- Range: \( (-\infty, \infty) \)
Since a logarithmic function can take any real-number value.
- x-intercept: \( (1, 0) \)
This occurs because \( \log_2(1) = 0 \).
- Vertical Asymptote: \( x = 0 \)
As \( x \) approaches zero from the right, \( \log_2 x \) approaches negative infinity.
2. Transformation to \( g(x) = \log_2(x + 4) + 8 \):
Let’s apply the transformations step by step.
- Horizontal Shift (left by 4 units):
\( g(x) = \log_2(x + 4) + 8 \)
This means replacing \( x \) with \( x + 4 \). The entire graph of \( f(x) \) shifts 4 units to the left.
- Vertical Shift (upward by 8 units):
\( g(x) = \log_2(x + 4) + 8 \)
This means the entire graph of \(f(x)\) will shift upward by 8 units.
3. Features of Transformed Function \( g(x) = \log_2(x + 4) + 8 \):
- Domain:
The domain shifts left by 4 units, so the domain of \( g(x) \) is \( (-4, \infty) \).
- Range:
The range of a logarithmic function is all real numbers, and this range does not change with translation. Thus, the range remains \( (-\infty, \infty) \).
- x-intercept:
The x-intercept of \( g(x) \) can be found by setting \( g(x) = 0 \):
[tex]\[ \log_2(x + 4) + 8 = 0 \implies \log_2(x + 4) = -8. \][/tex]
Converting this to exponential form, we get:
[tex]\[ x + 4 = 2^{-8} \implies x + 4 = 0.00390625 \implies x = 0.00390625 - 4 \implies x = -3.99609375. \][/tex]
Thus, the x-intercept is \( (-3.99609375, 0) \).
- Vertical Asymptote:
The vertical asymptote will shift left by 4 units, from \( x = 0 \) to \( x = -4 \).
### Summary of the Features:
- Domain: \( (-4, \infty) \).
- Range: \( (-\infty, \infty) \).
- x-intercept: \( (-3.99609375, 0) \).
- Vertical Asymptote: \( x = -4 \).
These are the key features of the function [tex]\( g(x) = \log_2(x + 4) + 8 \)[/tex].
