Answer :
To understand the features of the function [tex]\( g(x) = f(x + 4) + 8 \)[/tex] given the provided information about [tex]\( g \)[/tex], we need to reverse the transformations and determine the corresponding features of the original function [tex]\( f(x) \)[/tex].
Given:
1. [tex]\( g(x) \)[/tex] has a [tex]\( y \)[/tex]-intercept at [tex]\( (0,10) \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is [tex]\( (4, \infty) \)[/tex].
3. The range of [tex]\( g(x) \)[/tex] is [tex]\( (8, \infty) \)[/tex].
4. [tex]\( g(x) \)[/tex] has a vertical asymptote at [tex]\( x = -4 \)[/tex].
5. [tex]\( g(x) \)[/tex] has an [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex].
### Step-by-Step Solution:
#### 1. Y-Intercept of [tex]\( f(x) \)[/tex]:
The [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] occurs at [tex]\( (0,10) \)[/tex]. For [tex]\( g(0) = f(4) + 8 = 10 \)[/tex].
Therefore, [tex]\( f(4)=10-8=2 \)[/tex].
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (0,2) \)[/tex].
#### 2. Domain of [tex]\( f(x) \)[/tex]:
The domain of [tex]\( g(x) \)[/tex] is [tex]\( (4, \infty) \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], [tex]\( x + 4 \)[/tex] must be within the domain of [tex]\( f \)[/tex] for all [tex]\( x \ge 4 \)[/tex].
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
#### 3. Range of [tex]\( f(x) \)[/tex]:
The range of [tex]\( g(x) \)[/tex] is [tex]\( (8, \infty) \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], the range of [tex]\( f(x) \)[/tex] must be [tex]\( (0, \infty) \)[/tex]. This is because [tex]\( f(x + 4) = g(x) - 8 \)[/tex], thus the range of [tex]\( f(x) \)[/tex] results in [tex]\( [g(x) - 8]\)[/tex] which translates [tex]\( 8 - 8 = 0 \)[/tex] shifting the range down by 8.
#### 4. Vertical Asymptote of [tex]\( f(x) \)[/tex]:
The vertical asymptote of [tex]\( g(x) \)[/tex] is at [tex]\( x = -4 \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], this corresponds to the vertical asymptote of [tex]\( f(z) \)[/tex] being [tex]\( z = x + 4 \)[/tex]. Therefore, [tex]\( x + 4 = -4 \implies x = -8 \)[/tex].
So, the vertical asymptote of [tex]\( f(x) \)[/tex] is at [tex]\( x = -8 \)[/tex].
#### 5. X-Intercept of [tex]\( f(x) \)[/tex]:
The [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] occurs at [tex]\( (1,0) \)[/tex].
Therefore, [tex]\( g(1) = f(5) + 8 = 0 \)[/tex].
Thus, [tex]\( f(5) = -8 \)[/tex].
The [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (5 - 4) = (1-4, 0-8) = (-3, -8) \)[/tex].
### Summary:
- [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]: [tex]\( (0, 2) \)[/tex]
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Vertical asymptote of [tex]\( f(x) \)[/tex]: [tex]\( x = -8 \)[/tex]
- [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex]: [tex]\( (-3, -8) \)[/tex]
So, the features of the function [tex]\( f \)[/tex] are clearly determined through these transformations and relationships.
Given:
1. [tex]\( g(x) \)[/tex] has a [tex]\( y \)[/tex]-intercept at [tex]\( (0,10) \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is [tex]\( (4, \infty) \)[/tex].
3. The range of [tex]\( g(x) \)[/tex] is [tex]\( (8, \infty) \)[/tex].
4. [tex]\( g(x) \)[/tex] has a vertical asymptote at [tex]\( x = -4 \)[/tex].
5. [tex]\( g(x) \)[/tex] has an [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex].
### Step-by-Step Solution:
#### 1. Y-Intercept of [tex]\( f(x) \)[/tex]:
The [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] occurs at [tex]\( (0,10) \)[/tex]. For [tex]\( g(0) = f(4) + 8 = 10 \)[/tex].
Therefore, [tex]\( f(4)=10-8=2 \)[/tex].
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (0,2) \)[/tex].
#### 2. Domain of [tex]\( f(x) \)[/tex]:
The domain of [tex]\( g(x) \)[/tex] is [tex]\( (4, \infty) \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], [tex]\( x + 4 \)[/tex] must be within the domain of [tex]\( f \)[/tex] for all [tex]\( x \ge 4 \)[/tex].
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
#### 3. Range of [tex]\( f(x) \)[/tex]:
The range of [tex]\( g(x) \)[/tex] is [tex]\( (8, \infty) \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], the range of [tex]\( f(x) \)[/tex] must be [tex]\( (0, \infty) \)[/tex]. This is because [tex]\( f(x + 4) = g(x) - 8 \)[/tex], thus the range of [tex]\( f(x) \)[/tex] results in [tex]\( [g(x) - 8]\)[/tex] which translates [tex]\( 8 - 8 = 0 \)[/tex] shifting the range down by 8.
#### 4. Vertical Asymptote of [tex]\( f(x) \)[/tex]:
The vertical asymptote of [tex]\( g(x) \)[/tex] is at [tex]\( x = -4 \)[/tex].
Since [tex]\( g(x) = f(x + 4) + 8 \)[/tex], this corresponds to the vertical asymptote of [tex]\( f(z) \)[/tex] being [tex]\( z = x + 4 \)[/tex]. Therefore, [tex]\( x + 4 = -4 \implies x = -8 \)[/tex].
So, the vertical asymptote of [tex]\( f(x) \)[/tex] is at [tex]\( x = -8 \)[/tex].
#### 5. X-Intercept of [tex]\( f(x) \)[/tex]:
The [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] occurs at [tex]\( (1,0) \)[/tex].
Therefore, [tex]\( g(1) = f(5) + 8 = 0 \)[/tex].
Thus, [tex]\( f(5) = -8 \)[/tex].
The [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (5 - 4) = (1-4, 0-8) = (-3, -8) \)[/tex].
### Summary:
- [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]: [tex]\( (0, 2) \)[/tex]
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Vertical asymptote of [tex]\( f(x) \)[/tex]: [tex]\( x = -8 \)[/tex]
- [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex]: [tex]\( (-3, -8) \)[/tex]
So, the features of the function [tex]\( f \)[/tex] are clearly determined through these transformations and relationships.
