Question
The function below shows the cost of a hamburger with different numbers of toppings(t): f(t)=1.90+1.40t
what is the independent variable
what is the dependent variable
what is the slope and its meaning
what is the y-intercept and its meaning
find domain
find range
what is the independent variable
what is the dependent variable
what is the slope and its meaning
what is the y-intercept and its meaning
find domain
find range
Answers
GPT-4o mini
Let's analyze the function given:
\[ f(t) = 1.90 + 1.40t \]
### Independent and Dependent Variables:
- **Independent Variable:** The independent variable is \( t \), which represents the number of toppings on the hamburger. It is the variable that you can control or change.
- **Dependent Variable:** The dependent variable is \( f(t) \), which represents the cost of the hamburger based on the number of toppings. It depends on the number of toppings chosen.
### Slope and Its Meaning:
- **Slope:** The slope of the function is \( 1.40 \). This indicates that for every additional topping added to the hamburger, the cost increases by $1.40.
### Y-Intercept and Its Meaning:
- **Y-Intercept:** The y-intercept of the function is \( 1.90 \). This means that when there are no toppings (\( t = 0 \)), the base cost of the hamburger is $1.90.
### Domain:
The domain of the function is the set of all possible values for \( t \). Since you cannot have a negative number of toppings, the domain is:
\[ \text{Domain: } t \geq 0 \]
In interval notation, this is expressed as:
\[ [0, \infty) \]
### Range:
The range of the function consists of all possible values of \( f(t) \). As \( t \) increases from 0 to infinity, the cost \( f(t) \) increases without bound. The minimum value of \( f(t) \) occurs when \( t = 0 \):
\[ f(0) = 1.90 \]
So the range of the function is:
\[ \text{Range: } f(t) \geq 1.90 \]
In interval notation, this is expressed as:
\[ [1.90, \infty) \]
### Summary:
- Independent variable: \( t \) (number of toppings)
- Dependent variable: \( f(t) \) (cost of the hamburger)
- Slope: \( 1.40 \) (cost per additional topping)
- Y-intercept: \( 1.90 \) (base cost of the hamburger)
- Domain: \( [0, \infty) \)
- Range: \( [1.90, \infty) \)
\[ f(t) = 1.90 + 1.40t \]
### Independent and Dependent Variables:
- **Independent Variable:** The independent variable is \( t \), which represents the number of toppings on the hamburger. It is the variable that you can control or change.
- **Dependent Variable:** The dependent variable is \( f(t) \), which represents the cost of the hamburger based on the number of toppings. It depends on the number of toppings chosen.
### Slope and Its Meaning:
- **Slope:** The slope of the function is \( 1.40 \). This indicates that for every additional topping added to the hamburger, the cost increases by $1.40.
### Y-Intercept and Its Meaning:
- **Y-Intercept:** The y-intercept of the function is \( 1.90 \). This means that when there are no toppings (\( t = 0 \)), the base cost of the hamburger is $1.90.
### Domain:
The domain of the function is the set of all possible values for \( t \). Since you cannot have a negative number of toppings, the domain is:
\[ \text{Domain: } t \geq 0 \]
In interval notation, this is expressed as:
\[ [0, \infty) \]
### Range:
The range of the function consists of all possible values of \( f(t) \). As \( t \) increases from 0 to infinity, the cost \( f(t) \) increases without bound. The minimum value of \( f(t) \) occurs when \( t = 0 \):
\[ f(0) = 1.90 \]
So the range of the function is:
\[ \text{Range: } f(t) \geq 1.90 \]
In interval notation, this is expressed as:
\[ [1.90, \infty) \]
### Summary:
- Independent variable: \( t \) (number of toppings)
- Dependent variable: \( f(t) \) (cost of the hamburger)
- Slope: \( 1.40 \) (cost per additional topping)
- Y-intercept: \( 1.90 \) (base cost of the hamburger)
- Domain: \( [0, \infty) \)
- Range: \( [1.90, \infty) \)