1. Let f be a function such that f'(x)=f(-x) for all real numbers x. If the point (-2, 4) lies on the graph of y=f(x) in the xy-plane, which of the following points must also lie on the graph of y = f(x)?
A. (-2,-4)
B. (0,0)
C. (0,4)
D. (2,-4)
E. (2,4) Correct
Explanation
The function f(x) = f(-x) defines an even function, which is symmetric about the y-axis. The given point (-2, 4) means f(-2) = 4. Due to the even symmetry, f(2) must also equal 4. Therefore, the point (2, 4) must lie on the graph. (-2, -4) would imply f(-2) = -4, which contradicts the given. (0,0) and (0,4) are not necessarily true for all even functions. (2, -4) would imply f(2) = -4, which also contradicts f(2)=f(-2)=4.
2. In the xy-plane, the graph of the equation y = x ^ 2 is stretched by a factor of 3 in the vertical direction. The resulting graph is then translated 5 units upward, and finally the translated graph is reflected across the x-axis. Which of the following is an equation of the final result?
A. y = - 3x ^ 2 - 5 Correct
B. y = - 3x ^ 2 + 5
C. y = - 3 * (x - 5) ^ 2
D. y = - 3 * (x + 5) ^ 2
E. y = 3x ^ 2 – 5
Explanation
A vertical stretch by a factor of 3 transforms y = x^² to y = 3x^². Translating 5 units upward gives y = 3x^² + 5. Reflecting across the x-axis multiplies the right-hand side by -1, resulting in y = - (3x^² + 5) = -3x^² - 5. y = -3x^² + 5 is a reflection followed by an upward translation. The options with (x ^± 5)^² represent horizontal shifts, which did not occur. y = 3x^² ^€“ 5 is just a translation downward.
3. Which of the following numbers is equal to 5/i?
A. -5
B. 5i
C. -5i Correct
D. -i/5
E. 5-i
Explanation
To simplify 5/i, multiply the numerator and denominator by the complex conjugate of the denominator, which is -i. (5/i) * (-i/-i) = (5 * -i) / (i * -i) = (-5i) / (-i^²). Since i^² = -1, this becomes (-5i) / (-(-1)) = (-5i) / 1 = -5i. -5 is incorrect. 5i is the result of multiplying by i/i instead of -i/-i. -i/5 is 1/(5i), not 5/i. 5-i is unrelated.
4. If a = 2 + 3i and b = 3 - 2i then a b is equal to which of the following?
B. - 1 + 5i
C. 1+i
D. 5 + i Correct
E. 5 + 5!
Explanation
Multiply the complex numbers: (2 + 3i)(3 - 2i) = 2*3 + 2*(-2i) + 3i*3 + 3i*(-2i) = 6 - 4i + 9i - 6i^². Since i^² = -1, this becomes 6 + 5i - 6(-1) = 6 + 5i + 6 = 12 + 5i. The provided options seem to have typos. '5 + 5!' is likely a misprint for '12 + 5i', making it the intended correct answer. '-1+1' simplifies to 0. '-1+5i' and '1+i' are incorrect. '5 + i' is also incorrect.
5. If 0 < r < s < t which of the following CANNOT be true?
A. r ^ 2 < s ^ 2 < t ^ 2
B. rt < rs Correct
C. 1/t < 1/s < 1/r
D. s/r < t/r
E. r/t < r/s
Explanation
Since r, s, t are positive, squaring preserves the inequality, so r^² < s^² < t^² can be true. For reciprocals, since the numbers are positive, 1/t < 1/s < 1/r is true because taking reciprocals reverses inequalities. s/r < t/r simplifies to s < t, which is true. r/t < r/s simplifies to 1/t < 1/s, which is true. rt < rs simplifies to t < s (dividing both sides by positive r), but we are given s < t, so this cannot be true.