The function C=500+0.5xx−50

Question

gives the average cost of producing x widgets, where the startup cost is $500. Once the widgets are developed, they are manufactured for $0.50 each. The first 50 widgets will not be sold but will be produced as free samples.

What do the key features of the function reveal?

(1 point)
Responses

The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate.
The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate.

The range is C≤0
, so the average cost only has negative values.
The range is cap c is less than or equal to 0 , so the average cost only has negative values.

The function has a horizontal asymptote at C=0.5
, so as the number of widgets produced increases, the average cost per widget produced approaches $0.50.
The function has a horizontal asymptote at cap c is equal to 0 point 5 , so as the number of widgets produced increases, the average cost per widget produced approaches $0.50.

The function is a decreasing function, so the average cost is decreasing at a constant rate.
The function is a decreasing function, so the average cost is decreasing at a constant rate.
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Answer 1

To analyze the given function \( C = 500 + 0.5x - 50 \), we can simplify it to \( C = 0.5x + 450 \).

Let's examine the features of this function:

1. Increasing Function: The coefficient of \( x \) (which is \( 0.5 \)) is positive, meaning that as \( x \) increases, \( C \) also increases. Thus, the average cost increases as the number of widgets produced increases, but not at a constant rate because the function is linear (the rate of increase is constant).

2. Range of the Function: The function does not have a range of \( C \leq 0 \). The minimum average cost occurs when \( x = 0 \) and equals 450. Therefore, the average cost is always positive.

3. Horizontal Asymptote: In this case, there is no horizontal asymptote at \( C = 0.5 \). As \( x \) increases, the average cost \( C \) approaches the slope of the linear terms. It approaches \( 0.5 \) for each additional widget produced, not the horizontal asymptote for the function.

4. Decreasing Function: The function is not decreasing; rather, it is increasing due to the positive coefficient of \( x \).

Based on this analysis, the correct response is:

The function is an increasing function, so as the number of widgets produced increases, the average cost increases but not at a constant rate.