To find the correct statements that represent the function of the yearly cost in dollars, [tex]\( y \)[/tex], based on [tex]\( x \)[/tex], the number of game tokens purchased for a member of the arcade, let's break down the problem step-by-step.
### Step 1: Understanding the Membership Cost
The initial cost for a yearly membership is \[tex]$60. Regardless of the number of tokens purchased, this cost is fixed and represents the starting point, or the \( y \)-intercept, of the function.
### Step 2: Cost of Tokens
Members can purchase game tokens at the rate of \$[/tex]1.00 per 10 tokens. This implies a token costs:
[tex]\[ \text{Cost per token} = \frac{1.00\text{ dollar}}{10\text{ tokens}} = 0.10 \text{ dollars per token} \][/tex]
### Step 3: Constructing the Function
Considering the yearly membership cost and the cost per token:
[tex]\[ y = \text{Membership cost} + (\text{Cost per token} \times \text{Number of tokens}) \][/tex]
[tex]\[ y = 60 + 0.10 \times x \][/tex]
Simplified, this can be written as:
[tex]\[ y = \frac{1}{10} x + 60 \][/tex]
### Step 4: Identifying Key Features
Based on this function:
1. Slope: The coefficient of [tex]\( x \)[/tex] is [tex]\(\frac{1}{10}\)[/tex] or 0.1. This represents the cost per token.
2. [tex]\( y \)[/tex]-intercept: This is the constant term, which is 60 dollars, representing the yearly membership fee.
3. Domain: The number of tokens, [tex]\( x \)[/tex], must be non-negative. Therefore, the practical domain would be all non-negative real numbers.
4. Range: The minimum cost for a member is the membership cost itself, which is \[tex]$60. Any additional cost adds to this, so the range is \( y \mid y \geq 60 \).
### Correct Statements:
Based on this analysis, the correct answers are:
1. The \( y \)-intercept of the function is \(\$[/tex] 60\).
2. The function can be represented by the equation [tex]\( y = \frac{1}{10} x + 60 \)[/tex].
3. The range is [tex]\( \{ y \mid y \geq 60 \} \)[/tex].
### Incorrect Statements:
1. The statement "The slope of the function is \$1.00" is incorrect because the slope should be [tex]\( \frac{1}{10} \)[/tex] or 0.1, not 1.00.
2. The domain of the function isn't all real numbers; it would realistically only include non-negative values, thus making this statement imprecise.
Thus, the selected correct statements correctly describe the yearly cost function for a member based on the number of game tokens purchased.
