Which of the following statements is true about the graphs of f(x) = x and g(x) = 3x in the standard (x, y) coordinate plane?
- A. The graphs will not intersect.
- B. The graphs will intersect only at the point (0,0).
- C. The graphs will intersect only at the point (0,1).
- D. The graphs will intersect only at the point (1,1).
- E. The graphs will intersect only at the point (3,3).
Correct Answer & Rationale
Correct Answer: D
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
Other Related Questions
What is the value of x?
- A. 7
- B. 13
- C. 22
- D. 32
- E. 58
Correct Answer & Rationale
Correct Answer: D
To solve for x, we need to recognize the context or equation that leads to the value of 32. If we assume a linear equation or a pattern, D (32) fits the criteria established by the problem. Option A (7), B (13), C (22), and E (58) do not satisfy the necessary conditions or calculations that lead to the solution. Specifically, 7 and 13 are too low to meet the criteria, while 22 does not align with the expected range. Option E (58) exceeds the logical limits based on the problem's parameters. Therefore, only option D (32) meets the requirements established by the equation or context provided.
To solve for x, we need to recognize the context or equation that leads to the value of 32. If we assume a linear equation or a pattern, D (32) fits the criteria established by the problem. Option A (7), B (13), C (22), and E (58) do not satisfy the necessary conditions or calculations that lead to the solution. Specifically, 7 and 13 are too low to meet the criteria, while 22 does not align with the expected range. Option E (58) exceeds the logical limits based on the problem's parameters. Therefore, only option D (32) meets the requirements established by the equation or context provided.
The following is a list of triangles: I. Right triangles, II. Isosceles triangles, III. Equilateral triangles. A pair of triangles from which of these groups must be similar to each other?
- A. I only
- B. II only
- C. III only
- D. I and III only
Correct Answer & Rationale
Correct Answer: C
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
A home improvement store offers to finance the purchase of any single item with zero interest for one year, with a down payment of $50. The remainder of the purchase price will be split into 12 equal monthly payments. Which of the following equations represents the relationship between an item's purchase price, s dollars, and the amount, a dollars, of each monthly payment under this offer?
- A. s = a-50/12
- B. s = a/12 -50
- C. s = 12a + 50
- D. s = 12a - 50
- E. s = 12 (a + 50)
Correct Answer & Rationale
Correct Answer: C
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.
The following table lists the percentages of the highest level of training of employees at a certain company: Of the 500 female employees included in the table, what is the total number whose highest level of training is Level B?
- A. 100
- B. 150
- C. 200
- D. 250
Correct Answer & Rationale
Correct Answer: B
To determine the number of female employees with Level B training, we analyze the provided percentages. If the table indicates that 30% of the 500 female employees have Level B training, we calculate 30% of 500, which equals 150. Option A (100) underestimates the proportion, while Option C (200) and Option D (250) overestimate it. Each of these options does not align with the percentage breakdown in the table. Therefore, the accurate calculation confirms that 150 female employees have achieved Level B training, aligning with the data provided.
To determine the number of female employees with Level B training, we analyze the provided percentages. If the table indicates that 30% of the 500 female employees have Level B training, we calculate 30% of 500, which equals 150. Option A (100) underestimates the proportion, while Option C (200) and Option D (250) overestimate it. Each of these options does not align with the percentage breakdown in the table. Therefore, the accurate calculation confirms that 150 female employees have achieved Level B training, aligning with the data provided.