Let f(x) = 3x². What is f(-2x)?
- A. -36x²
- B. -12x²
- C. -6x²
- D. 12x²
- E. 36x²
Correct Answer & Rationale
Correct Answer: D
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
Other Related Questions
What are the solutions to the equation: x² - 10?
- A. ±5
- B. ±√10
- C. ±10
- D. ±10²
- E. ±20
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
sqrt(45) is between what two consecutive whole numbers?
- A. 4 and 5
- B. 5 and 6
- C. 6 and 7
- D. 14 and 15
- E. 22 and 23
Correct Answer & Rationale
Correct Answer: C
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
Isabel earns $15.80 per hour for the first 8 hours she works each day. She earns 1.5 times her hourly rate for all time after the first 8 hours. How much does Isabel earn on a day when she works 8.5 hours?
- A. 126.4
- B. 138.25
- C. 189.6
- D. 201.45
- E. 237
Correct Answer & Rationale
Correct Answer: B
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
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.