To analyze the function [tex]\( g(x) = f(x+4) + 8 \)[/tex], let's determine its key features step-by-step.
### 1. [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. So,
[tex]\[ g(0) = f(0 + 4) + 8 = f(4) + 8 \][/tex]
Given:
\[ y \)-intercept at [tex]\( (0, 10) \\
Thus, \( g(0) = 10 \)[/tex] implies:
\[ f(4) + 8 = 10 \\
f(4) = 2 \)
### 2. Range
The range of [tex]\( g(x) \)[/tex] is determined by the minimum value of [tex]\( f(x) \)[/tex] shifted upwards by 8 units. Given the range of [tex]\( (8, \infty) \)[/tex]:
\[ g(x) \geq 8 \\
Thus, [tex]\( f(x+4) \geq 0 \)[/tex]
### 3. Domain
The domain of [tex]\( g(x) \)[/tex] is shifted compared to [tex]\( f(x) \)[/tex]. Given the domain [tex]\( (4, \infty) \)[/tex]:
\[ x + 4 >0\\
So, [tex]\( x > -4 \
\Thus, x > 4 \)[/tex]
### 4. [tex]\( x \)[/tex]-intercept
The [tex]\( x \)[/tex]-intercept occurs where [tex]\( g(x) = 0 \)[/tex]. Thus,
\[ 0 = f(x+4) + 8\\
f(x+4) = -8 \\
If [tex]\( x = 1 \)[/tex], then:
\Thus,
\Thus, f(x+4\\)
### 5. Vertical Asymptote
The vertical asymptote occurs where [tex]\( g(x) \)[/tex] approaches infinity as [tex]\( x \)[/tex] approaches certain value [tex]\( c \)[/tex]. Given vertical asymptote at:
\[ x = -4 \) implies that:
\[ f(x + 4) \) undefined at [tex]\( x+4 = x = -4 \So]\( x + 4 = 0 \
\Thus,
### Conclusion
We have the following features of the function \( g(x) \)[/tex]:
1. [tex]\( y \)[/tex]-intercept: [tex]\( (0, 10) \)[/tex]
2. Range: [tex]\( (8, \infty) \)[/tex]
3. Domain: [tex]\( (4, \infty) \)[/tex]
4. [tex]\( x \)[/tex]-intercept: [tex]\( (1, \)[/tex]
5. Vertical asymptote: \( x = -4
