Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
