Question 3

Question

A)
Consider the following verbal description of a rational function. Where are the asymptotes and point of discontinuity located for the described rational function?

f of x equals the quotient of x plus four and x squared minus sixteen

(1 point)
Responses

vertical asymptote: x=4

horizontal asymptote: y=12

point of discontinuity: none

vertical asymptote: x is equal to 4 horizontal asymptote: y is equal to 1 half point of discontinuity: none

vertical asymptote: x=4

horizontal asymptote: y=0

point of discontinuity: when x=−4

vertical asymptote: x is equal to 4 horizontal asymptote: y is equal to 0 point of discontinuity: when x is equal to negative 4

vertical asymptote: y=0

horizontal asymptote: x=4

point of discontinuity: when x=−4

vertical asymptote: y is equal to 0 horizontal asymptote: x is equal to 4 point of discontinuity: when x is equal to negative 4

vertical asymptote: x=4

horizontal asymptote: none

point of discontinuity: none

vertical asymptote: x is equal to 4 horizontal asymptote: none point of discontinuity: none
Question 4
A)
The function C=320+x can be used to find the concentration of fertilizer in a mixture that contains a concentration of fertilizer of 15% for 20 ounces of water when x ounces of water are added to the mixture.

What do the key features of the function reveal?

(1 point)
Responses

The function is a decreasing function, so as the number of ounces of water added to the mixture increases, the concentration of fertilizer decreases but not at a constant rate.
The function is a decreasing function, so as the number of ounces of water added to the mixture increases, the concentration of fertilizer decreases but not at a constant rate.

The function is an increasing function, so as the number of ounces of water added to the mixture increases, the concentration of fertilizer increases but not at a constant rate.
The function is an increasing function, so as the number of ounces of water added to the mixture increases, the concentration of fertilizer increases but not at a constant rate.

There is a horizontal asymptote at C=0 which indicates that when the amount of water added to the mixture increases, the concentration of fertilizer is 0%.
There is a horizontal asymptote at cap c is equal to 0 which indicates that when the amount of water added to the mixture increases, the concentration of fertilizer is 0%.

The domain is x≤0, so only negative amounts of water are being added to the mixture.
The domain is x is less than or equal to 0 , so only negative amounts of water are being added to the mixture.
Question 5
A)
The function C=500+0.5xx−50 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 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.

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.

Answer 1

Let's address each question one by one based on the information provided.

Question 3

We need to analyze the rational function given: \[ f(x) = \frac{x + 4}{x^2 - 16} \]

1. Vertical Asymptotes: These occur where the denominator equals zero (and the numerator is not zero at those points).

   • The denominator \( x^2 - 16 = 0 \) gives: \[ x^2 = 16 \implies x = \pm 4 \]
   • This means vertical asymptotes at \( x = 4 \) and \( x = -4 \).

2. Horizontal Asymptotes: For rational functions, we observe the degrees of the numerator and denominator:

   • Degree of numerator (1) < degree of denominator (2). Thus, the horizontal asymptote is at \( y = 0 \).

3. Point of Discontinuity: This occurs where the function is undefined, which is at \( x = 4 \) and \( x = -4 \).

So, the correct choice is:

• Vertical asymptote: \( x = 4 \)
• Horizontal asymptote: \( y = 0 \)
• Point of discontinuity: when \( x = -4 \)

Question 4

Given the function: \[ C = \frac{320}{x} + 15 \] We analyze the behavior associated with adding water in the mixture.

1. Increasing or Decreasing: The function represents the concentration of fertilizer as you add more water. As you add more water (increasing \( x \)), the concentration \( C \) decreases (since \( C \) is dependent on \( \frac{320}{x} \)), approaching 0% but never reaching it since you are adding more water to the same amount of fertilizer.

2. Horizontal Asymptote: As the amount of water increases indefinitely, the fraction \( \frac{320}{x} \) approaches 0, leading to a concentration around 15%, but since we are adding more water, we observe that the concentration goes to less than 15%.

3. Domain: \( x \) cannot be negative because we cannot add negative ounces of water.

Therefore, the correct choice is:

• The function is a decreasing function, so as the number of ounces of water added to the mixture increases, the concentration of fertilizer decreases but not at a constant rate.

Question 5

Given the function: \[ C = \frac{500 + 0.5x}{x - 50} \]

1. Horizontal Asymptote: To find the behavior as \( x \) becomes very large:

   • The leading coefficient of the numerator is \( 0.5 \) and for the denominator, it's 1. Therefore, the horizontal asymptote is at \( y = 0.5 \) as \( x \to \infty \).

2. Increasing or Decreasing: As \( x \) increases, mainly after reaching the startup cost, the average cost per widget initially decreases and then stabilizes as production increases, due to the fixed cost being spread over an increasing number of items.

3. Range: The range of \( C \) does not contain negative values since average cost cannot be negative.

Thus, the correct choice is:

• 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.

If you have any further questions or need additional explanations, feel free to ask!