Answer :
Let's delve into the given problem step-by-step to identify which of the functions represent exponential growth and which have a horizontal asymptote.
### Step 1: Identify Exponential Growth Functions
We need to determine which of the functions listed exhibits an exponential growth pattern. Exponential growth typically happens when a quantity increases by a constant factor over equal increments of time or another variable.
Given the provided information, the functions that exhibit exponential growth are:
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
These functions are marked with a check (✓) in the problem statement indicating that they represent exponential growth.
### Step 2: Identify Functions with a Horizontal Asymptote
Next, we need to identify which functions have a horizontal asymptote. A horizontal asymptote is a horizontal line that the graph of the function approaches as [tex]\( x \)[/tex] (or the other variable) goes to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
Given the provided information, the functions that have a horizontal asymptote are:
- [tex]\( y = f(x) \)[/tex]
- [tex]\( y = g(x) \)[/tex]
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
All these functions are listed to have a horizontal asymptote.
### Summary:
- Exponential Growth Functions:
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
- Functions with a Horizontal Asymptote:
- [tex]\( y = f(x) \)[/tex]
- [tex]\( y = g(x) \)[/tex]
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
In summary, based on our analysis:
- [tex]\( y = h(x) \)[/tex] and [tex]\( y = k(x) \)[/tex] represent exponential growth.
- [tex]\( y = f(x) \)[/tex], [tex]\( y = g(x) \)[/tex], [tex]\( y = h(x) \)[/tex], and [tex]\( y = k(x) \)[/tex] have horizontal asymptotes.
### Step 1: Identify Exponential Growth Functions
We need to determine which of the functions listed exhibits an exponential growth pattern. Exponential growth typically happens when a quantity increases by a constant factor over equal increments of time or another variable.
Given the provided information, the functions that exhibit exponential growth are:
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
These functions are marked with a check (✓) in the problem statement indicating that they represent exponential growth.
### Step 2: Identify Functions with a Horizontal Asymptote
Next, we need to identify which functions have a horizontal asymptote. A horizontal asymptote is a horizontal line that the graph of the function approaches as [tex]\( x \)[/tex] (or the other variable) goes to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
Given the provided information, the functions that have a horizontal asymptote are:
- [tex]\( y = f(x) \)[/tex]
- [tex]\( y = g(x) \)[/tex]
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
All these functions are listed to have a horizontal asymptote.
### Summary:
- Exponential Growth Functions:
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
- Functions with a Horizontal Asymptote:
- [tex]\( y = f(x) \)[/tex]
- [tex]\( y = g(x) \)[/tex]
- [tex]\( y = h(x) \)[/tex]
- [tex]\( y = k(x) \)[/tex]
In summary, based on our analysis:
- [tex]\( y = h(x) \)[/tex] and [tex]\( y = k(x) \)[/tex] represent exponential growth.
- [tex]\( y = f(x) \)[/tex], [tex]\( y = g(x) \)[/tex], [tex]\( y = h(x) \)[/tex], and [tex]\( y = k(x) \)[/tex] have horizontal asymptotes.
