Answer :
To determine which functions have the given key features, we need to analyze those features step by step. Let's look at each feature with respect to the given functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex].
1. [tex]\(y\)[/tex]-intercept at [tex]\((0, 4)\)[/tex]:
- For a function to have a [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex], it must pass through the point [tex]\((0, 4)\)[/tex]. We need to check [tex]\( f(0) \)[/tex], [tex]\( g(0) \)[/tex], and [tex]\( h(0) \)[/tex] to determine if any of them equals 4.
2. [tex]\(x\)[/tex]-intercept at [tex]\((1, 0)\)[/tex]:
- Here, we need to identify if [tex]\( f(1) \)[/tex], [tex]\( g(1) \)[/tex], or [tex]\( h(1) \)[/tex] equates to 0. This implies we're seeking if the functions are zero at [tex]\( x = 1 \)[/tex].
3. Increasing on all intervals of [tex]\( x \)[/tex]:
- For a function to be increasing on all intervals of [tex]\( x \)[/tex], its first derivative must be positive for all [tex]\( x \)[/tex]. We will consider which of the functions exhibit this behavior over their entire domain.
4. Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
- Here, we're analyzing the end behavior of the functions as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]. Specifically, we need to determine if the limit of the functions as [tex]\( x \to -\infty \)[/tex] is an integer.
### Solution Steps:
1. [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex]:
- If [tex]\( y = 4 \)[/tex] when [tex]\( x = 0 \)[/tex], we have:
- [tex]\( f(0) = 4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex]
- [tex]\( h(0) = 4 \)[/tex]
Given that this point is for the [tex]\( y \)[/tex]-intercept at [tex]\( (0, 4) \)[/tex], let's fill the information accordingly:
[tex]\[ \boxed{f(x) \text{ and } g(x)} \][/tex]
2. [tex]\( x \)[/tex]-intercept at [tex]\((1, 0)\)[/tex]:
- If [tex]\( x = 1 \)[/tex] makes [tex]\( y = 0 \)[/tex], we have:
- [tex]\( f(1) = 0 \)[/tex]
- [tex]\( g(1) = 0 \)[/tex]
- [tex]\( h(1) = 0 \)[/tex]
Given that this point is for the [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex], we mark:
[tex]\[ \boxed{\text{all three functions}} \][/tex]
3. Increasing on all intervals of [tex]\( x \)[/tex]:
- Check to see which functions are always increasing:
- [tex]\( f(x) \)[/tex] is increasing on all intervals
- [tex]\( g(x) \)[/tex] is increasing on all intervals
- [tex]\( h(x) \)[/tex] is increasing on all intervals
Given that all functions are increasing on all intervals, mark:
[tex]\[ \boxed{\text{all three functions}} \][/tex]
4. Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]:
- We examine the behavior as [tex]\( x \to -\infty \)[/tex]:
- [tex]\( f(x) \to \text{an integer as } x \to -\infty \)[/tex]
- [tex]\( g(x) \to \text{an integer as } x \to -\infty \)[/tex]
- [tex]\( h(x) \to \text{an integer as } x \to -\infty \)[/tex]
Since only [tex]\( h(x) \)[/tex] approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], we mark:
[tex]\[ \boxed{h(x)} \][/tex]
Combining all observations:
- [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex]: \boxed{g(x)}
- [tex]\( x \)[/tex]-intercept at [tex]\((1, 0)\)[/tex]: \boxed{\text{all three functions}}
- Increasing on all intervals of [tex]\( x \)[/tex]: \boxed{\text{all three functions}}
- Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]: \boxed{h(x)}
Thus, we have comprehensively matched the key features to the given functions accordingly.
1. [tex]\(y\)[/tex]-intercept at [tex]\((0, 4)\)[/tex]:
- For a function to have a [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex], it must pass through the point [tex]\((0, 4)\)[/tex]. We need to check [tex]\( f(0) \)[/tex], [tex]\( g(0) \)[/tex], and [tex]\( h(0) \)[/tex] to determine if any of them equals 4.
2. [tex]\(x\)[/tex]-intercept at [tex]\((1, 0)\)[/tex]:
- Here, we need to identify if [tex]\( f(1) \)[/tex], [tex]\( g(1) \)[/tex], or [tex]\( h(1) \)[/tex] equates to 0. This implies we're seeking if the functions are zero at [tex]\( x = 1 \)[/tex].
3. Increasing on all intervals of [tex]\( x \)[/tex]:
- For a function to be increasing on all intervals of [tex]\( x \)[/tex], its first derivative must be positive for all [tex]\( x \)[/tex]. We will consider which of the functions exhibit this behavior over their entire domain.
4. Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
- Here, we're analyzing the end behavior of the functions as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]. Specifically, we need to determine if the limit of the functions as [tex]\( x \to -\infty \)[/tex] is an integer.
### Solution Steps:
1. [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex]:
- If [tex]\( y = 4 \)[/tex] when [tex]\( x = 0 \)[/tex], we have:
- [tex]\( f(0) = 4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex]
- [tex]\( h(0) = 4 \)[/tex]
Given that this point is for the [tex]\( y \)[/tex]-intercept at [tex]\( (0, 4) \)[/tex], let's fill the information accordingly:
[tex]\[ \boxed{f(x) \text{ and } g(x)} \][/tex]
2. [tex]\( x \)[/tex]-intercept at [tex]\((1, 0)\)[/tex]:
- If [tex]\( x = 1 \)[/tex] makes [tex]\( y = 0 \)[/tex], we have:
- [tex]\( f(1) = 0 \)[/tex]
- [tex]\( g(1) = 0 \)[/tex]
- [tex]\( h(1) = 0 \)[/tex]
Given that this point is for the [tex]\( x \)[/tex]-intercept at [tex]\( (1, 0) \)[/tex], we mark:
[tex]\[ \boxed{\text{all three functions}} \][/tex]
3. Increasing on all intervals of [tex]\( x \)[/tex]:
- Check to see which functions are always increasing:
- [tex]\( f(x) \)[/tex] is increasing on all intervals
- [tex]\( g(x) \)[/tex] is increasing on all intervals
- [tex]\( h(x) \)[/tex] is increasing on all intervals
Given that all functions are increasing on all intervals, mark:
[tex]\[ \boxed{\text{all three functions}} \][/tex]
4. Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]:
- We examine the behavior as [tex]\( x \to -\infty \)[/tex]:
- [tex]\( f(x) \to \text{an integer as } x \to -\infty \)[/tex]
- [tex]\( g(x) \to \text{an integer as } x \to -\infty \)[/tex]
- [tex]\( h(x) \to \text{an integer as } x \to -\infty \)[/tex]
Since only [tex]\( h(x) \)[/tex] approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], we mark:
[tex]\[ \boxed{h(x)} \][/tex]
Combining all observations:
- [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex]: \boxed{g(x)}
- [tex]\( x \)[/tex]-intercept at [tex]\((1, 0)\)[/tex]: \boxed{\text{all three functions}}
- Increasing on all intervals of [tex]\( x \)[/tex]: \boxed{\text{all three functions}}
- Approaches an integer as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]: \boxed{h(x)}
Thus, we have comprehensively matched the key features to the given functions accordingly.
