Question
A video game arcade offers a yearly membership with reduced rates for game play. A single membership costs $60 per year. Game tokens can be purchased by members at the reduced rate of $1.00 per 10 tokens.
Which statements represent the function of yearly cost in dollars, y, based on x, the number of game tokens purchased for a member of the arcade? Select the three that apply.
(3 points)
Responses
The slope of the function is $1
The slope of the function is $1
The domain of the function is all real numbers.
The domain of the function is all real numbers.
The function can be represented by the equation y=110x+60
The function can be represented by the equation y is equal to 1 tenth x plus 60
The y-intercept of the function is $60.
The y-intercept of the function is $60.
The range of the function is y≥60
.
Which statements represent the function of yearly cost in dollars, y, based on x, the number of game tokens purchased for a member of the arcade? Select the three that apply.
(3 points)
Responses
The slope of the function is $1
The slope of the function is $1
The domain of the function is all real numbers.
The domain of the function is all real numbers.
The function can be represented by the equation y=110x+60
The function can be represented by the equation y is equal to 1 tenth x plus 60
The y-intercept of the function is $60.
The y-intercept of the function is $60.
The range of the function is y≥60
.
Answers
GPT-4o mini
To determine the proper statements about the function representing the yearly cost \( y \) based on the number of game tokens \( x \) purchased by a member of the arcade, we will analyze the membership cost, the cost of tokens, and the structure of the function.
1. **Membership Cost**: The yearly membership costs $60.
2. **Cost of Tokens**: Members can purchase tokens at a rate of $1.00 for 10 tokens, or \( \frac{1}{10} \) dollars per token.
### Constructing the Cost Function
The cost function can be expressed as:
\[
y = 60 + \left(\frac{1}{10}x\right)
\]
where:
- \( 60 \) is the fixed cost (y-intercept),
- \( \frac{1}{10}x \) represents the cost of buying \( x \) tokens.
### Analyzing the Statements:
- **The slope of the function is $1**: **False**
- The slope is actually \( \frac{1}{10} \), not $1.
- **The domain of the function is all real numbers**: **False**
- The domain is limited to non-negative integers (no negative tokens).
- **The function can be represented by the equation \( y = 0.1x + 60 \)**: **True**
- This correctly represents the cost function with \( \frac{1}{10} \) represented as \( 0.1 \).
- **The y-intercept of the function is $60**: **True**
- The y-intercept (when \( x = 0 \)) is indeed $60.
- **The range of the function is \( y \geq 60 \)**: **True**
- Since the least cost (just for the membership without tokens) is $60, the range starts at $60 and increases with more tokens.
### Correct Statements
Therefore, the three applicable statements are:
1. The function can be represented by the equation \( y = 0.1x + 60 \).
2. The y-intercept of the function is $60.
3. The range of the function is \( y \geq 60 \).
1. **Membership Cost**: The yearly membership costs $60.
2. **Cost of Tokens**: Members can purchase tokens at a rate of $1.00 for 10 tokens, or \( \frac{1}{10} \) dollars per token.
### Constructing the Cost Function
The cost function can be expressed as:
\[
y = 60 + \left(\frac{1}{10}x\right)
\]
where:
- \( 60 \) is the fixed cost (y-intercept),
- \( \frac{1}{10}x \) represents the cost of buying \( x \) tokens.
### Analyzing the Statements:
- **The slope of the function is $1**: **False**
- The slope is actually \( \frac{1}{10} \), not $1.
- **The domain of the function is all real numbers**: **False**
- The domain is limited to non-negative integers (no negative tokens).
- **The function can be represented by the equation \( y = 0.1x + 60 \)**: **True**
- This correctly represents the cost function with \( \frac{1}{10} \) represented as \( 0.1 \).
- **The y-intercept of the function is $60**: **True**
- The y-intercept (when \( x = 0 \)) is indeed $60.
- **The range of the function is \( y \geq 60 \)**: **True**
- Since the least cost (just for the membership without tokens) is $60, the range starts at $60 and increases with more tokens.
### Correct Statements
Therefore, the three applicable statements are:
1. The function can be represented by the equation \( y = 0.1x + 60 \).
2. The y-intercept of the function is $60.
3. The range of the function is \( y \geq 60 \).