Let g(x) = x². What is the average rate of change of the function from x = 4 to x = 8?
- A. 1/12
- B. $2
- C. $4
- D. $12
- E. $48
Correct Answer & Rationale
Correct Answer: C
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
Other Related Questions
Which of the following equations does not represent y as a function of x in the standard (x, y) coordinate plane?
- A. y = x
- B. y = x + 2
- C. y = x² + 2
- D. x = y + 2
- E. x = y² + 2
Correct Answer & Rationale
Correct Answer: E
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
The number of years the employee has been employed by the city is at least 25 years. The sum of the employee's age and number of years employed by the city is at least 90 years. Larry has been employed by the city since his 38th birthday. Assuming he continues to work for the city, at what age will he first qualify for full retirement benefits?
- A. 52
- B. 55
- C. 62
- D. 63
- E. 64
Correct Answer & Rationale
Correct Answer: E
To qualify for full retirement benefits, Larry must be at least 25 years employed and have a combined age and years of service of at least 90 years. Since he started working at age 38, he will reach 25 years of employment at age 63. At that point, his age (63) plus his years of service (25) totals 88, which does not meet the 90-year requirement. At age 64, he will have 26 years of service, bringing the total to 90 years (64 + 26), thus meeting both criteria. Options A (52), B (55), and C (62) do not allow for 25 years of service, while D (63) fails to meet the age and service sum requirement.
To qualify for full retirement benefits, Larry must be at least 25 years employed and have a combined age and years of service of at least 90 years. Since he started working at age 38, he will reach 25 years of employment at age 63. At that point, his age (63) plus his years of service (25) totals 88, which does not meet the 90-year requirement. At age 64, he will have 26 years of service, bringing the total to 90 years (64 + 26), thus meeting both criteria. Options A (52), B (55), and C (62) do not allow for 25 years of service, while D (63) fails to meet the age and service sum requirement.
In a survey of 300 people who were randomly sampled from a well-defined population, 60 said that they read a newspaper daily. If 1,000 people had been randomly sampled from the same population and asked the same question, how many would be expected to say they read a newspaper daily?
- A. 180
- B. 200
- C. 360
- D. 600
- E. 760
Correct Answer & Rationale
Correct Answer: A
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
Each month, the charge for a lawn care service consists of a flat fee of $25, plus $5 each time the lawn is mowed. Which of the following equations represents the total monthly charge, A(m), in dollars, as a function of the number of times the lawn is mowed, m?
- A. A(m) = 5(25)m
- B. A(m) = 5 + 25m
- C. A(m) = 5m + 25
- D. A(m) = 25m + 5
- E. A(m) = m + 5 + 25
Correct Answer & Rationale
Correct Answer: C
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.