Choose the best answer. If necessary, use the paper you were given.
If the function g is defined by g (x) = x/(x+1)', which of the following is true?
- A. g (10) <g (20)
- B. g (20) <g (10)
- C. g(0) =1
- D. g(1)=0
Correct Answer & Rationale
Correct Answer: A
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.
Other Related Questions
The expressions x - 2 and x + 3 represent the length and width of a rectangle, respectively. If the area of the rectangle is 24, what is the perimeter of the rectangle?
- A. 20
- B. 22
- C. 24
- D. 28
Correct Answer & Rationale
Correct Answer: B
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
If an item regularly costs d dollars and is discounted 12 percent, which of the following represents the discounted price in dollars?
- A. 0.12d
- B. 0.88d
- C. 1.12d
- D. d-0.12
Correct Answer & Rationale
Correct Answer: B
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
A shirt is on sale for 15 percent off the original price of x dollars. If a customer has a coupon for 5 dollars off the sale price, which of the following represents the price, in dollars, the customer will pay, excluding tax, for the shirt?
- A. 0.15x-5
- B. 0.85x -5
- C. 0.85(x-5)
- D. 5-0.85x
Correct Answer & Rationale
Correct Answer: B
To determine the price a customer pays after applying both discounts, start with the original price, x. A 15% discount reduces the price to 85% of the original, calculated as 0.85x. After this, the customer applies a $5 coupon, leading to the final price of 0.85x - 5. Option A (0.15x - 5) incorrectly calculates the discount as a direct subtraction from the original price, misrepresenting the order of operations. Option C (0.85(x - 5)) mistakenly applies the coupon before calculating the discount, which is not the correct sequence. Option D (5 - 0.85x) suggests a negative price, which is nonsensical in this context.
To determine the price a customer pays after applying both discounts, start with the original price, x. A 15% discount reduces the price to 85% of the original, calculated as 0.85x. After this, the customer applies a $5 coupon, leading to the final price of 0.85x - 5. Option A (0.15x - 5) incorrectly calculates the discount as a direct subtraction from the original price, misrepresenting the order of operations. Option C (0.85(x - 5)) mistakenly applies the coupon before calculating the discount, which is not the correct sequence. Option D (5 - 0.85x) suggests a negative price, which is nonsensical in this context.
The sum of n and the product 3 times n is 12. What is the value of n?
- A. 2
- B. 3
- C. 4
- D. 4 ½
Correct Answer & Rationale
Correct Answer: B
To solve the equation formed by the problem statement, we express it as \( n + 3n = 12 \), which simplifies to \( 4n = 12 \). Dividing both sides by 4 gives \( n = 3 \). Option A (2) does not satisfy the equation, as substituting it results in \( 2 + 6 = 8 \), which is incorrect. Option C (4) leads to \( 4 + 12 = 16 \), also incorrect. Option D (4 ½) results in \( 4.5 + 13.5 = 18 \), which is too high. Thus, only \( n = 3 \) fulfills the original equation, confirming its validity.
To solve the equation formed by the problem statement, we express it as \( n + 3n = 12 \), which simplifies to \( 4n = 12 \). Dividing both sides by 4 gives \( n = 3 \). Option A (2) does not satisfy the equation, as substituting it results in \( 2 + 6 = 8 \), which is incorrect. Option C (4) leads to \( 4 + 12 = 16 \), also incorrect. Option D (4 ½) results in \( 4.5 + 13.5 = 18 \), which is too high. Thus, only \( n = 3 \) fulfills the original equation, confirming its validity.