Question

jack rents a space at a farmer's market and pays the market manger $6.50 for each gourd he sells. the amount jack pays each day is given f(x)=6.50x where x is the gourds sold, graph this function and use set-builder and interval notation to express the domain and range.

Answer 1

To graph the function \(f(x) = 6.50x\), which represents the amount Jack pays based on the number of gourds sold, and express its domain and range using set-builder and interval notation:

1. Graph the function:

- The x-axis represents the number of gourds sold (x).

- The y-axis represents the amount paid (f(x)).

The graph is a straight line with a slope of $6.50, passing through the origin (0,0). As x (number of gourds sold) increases, the amount paid also increases linearly.

2. Domain:

- In this context, the domain represents the possible values for the number of gourds sold.

- Since you can sell any non-negative number of gourds, the domain is all non-negative real numbers.

- In interval notation: \([0, \infty)\)

- In set-builder notation: \(\{x \mid x \geq 0\}\)

3. Range:

- The range represents the possible values for the amount paid.

- Since the amount paid is directly proportional to the number of gourds sold, it can take any non-negative real value.

- In interval notation: \([0, \infty)\)

- In set-builder notation: \(\{f(x) \mid f(x) \geq 0\}\) or simply \(\mathbb{R}^+\) to denote all positive real numbers.

So, the graph is a line that starts at the origin, and its domain and range both extend to all non-negative real numbers.

Answer 2

The function f(x)=6.50x is graphed by plotting points starting from the origin and shows the amount paid to the market manager based on gourds sold. The domain is all non-negative integers, and the range is all non-negative multiples of $6.50.

Step-by-step explanation:

The function given, f(x)=6.50x, represents the amount Jack pays the market manager at a farmer's market for each gourd he sells. To graph this function, plot the y-intercept at the origin (0,0) because there is no fixed cost and then plot another point like (1, 6.50) which represents selling one gourd for $6.50. Connect these two points with a straight line extending indefinitely to the right, as there is theoretically no limit to the number of gourds Jack can sell.

The domain of this function is the set of all non-negative integers, since you cannot sell a negative number of gourds and selling fractions of gourds is not typically practical. In set-builder notation, this is x . In interval notation, the domain is [0, ∞). The range is the set of all non-negative multiples of $6.50, which in set-builder notation is 6.50x and in interval notation is [0, ∞).