Since g is an even function, g(−4) = g(4) = 8.

Alternatively: First find the value of a, and then find g(−4). Since g(4) = 8,substituting 4 for x and 8 for g(x) gives 8 = a(4)² + 24 = 16a + 24. Solving this last equation gives a = −1. Thus g(x) = −x² + 24, from which it follows that g(−4) = −(−4)² + 24; g(−4) = −16 + 24; and g(−4) = 8.

Choices B, C, and D are incorrect because g is a function and there can only be one value of g(−4).

Adding x and 19 to both sides of 2y − x = −19 gives x = 2y + 19. Then, substituting 2y + 19 for x in 3x + 4y = −23 gives 3(2y + 19) + 4y = −23. This last equation is equivalent to 10y + 57 = −23. Solving 10y + 57 = −23 gives y = −8. Finally, substituting −8 for y in 2y − x = −19 gives 2(−8) − x = −19, or x = 3. Therefore, the solution (x, y) to the given system of equations is (3, −8).

Choices A, C, and D are incorrect because when the given values of x and y are substituted in 2y − x = −19, the value of the left side of the equation does not equal −19.

Since ^a⁄_b= 2, it follows that ^b⁄_a = ¹⁄₂. Multiplying both sides of the equation by 4 gives 4 (^b⁄_a) = \(\frac{4b}{a}\) = 2.

Choice A is incorrect because if \(\frac{4b}{a}\) = 0, then ^a⁄_b would be undefined. Choice B is incorrect because if \(\frac{4b}{a}\) = 1, then ^a⁄_b = 4. Choice D is incorrect because if \(\frac{4b}{a}\) = 4, then ^a⁄_b = 1.

Since the right-hand side of the equation is P times the expression \(\frac{\left ( \frac{r}{1,200} \right )\left ( 1+\frac{r}{1,200} \right )^{N}}{\left ( 1+\frac{r}{1,200} \right )^{N}-1}\), multiplying both sides of the equation by the reciprocal of this expression results in \(\frac{\left ( 1+\frac{r}{1,200} \right )^{N}-1}{\left ( \frac{r}{1,200} \right )\left ( 1+\frac{r}{1,200} \right )^{N}}m=P\)

Choices A, C, and D are incorrect and are likely the result of conceptual or computation errors while trying to solve for P.

In the equation h = 3a + 28.6, if a, the age of the boy, increases by 1, then h becomes h = 3(a + 1) + 28.6 = 3a + 3 + 28.6 = (3a + 28.6) + 3. Therefore, the model estimates that the boy’s height increases by 3 inches each year.

Alternatively: The height, h, is a linear function of the age, a, of the boy. The coefficient 3 can be interpreted as the rate of change of the function; in this case, the rate of change can be described as a change of 3 inches in height for every additional year in age.

Choices B, C, and D are incorrect and are likely to result from common errors in calculating the value of h or in calculating the difference between the values of h for different values of a.

Only like terms, with the same variables and exponents, can be combined to determine the answer as shown here:

(x²y − 3y² + 5xy²) − (−x²y + 3xy² − 3y²)

= (x²y − (−x²y)) + (−3y² − (−3y²)) + (5xy² − 3xy²)

= 2x²y + 0 + 2xy²

= 2x²y + 2xy²

Choices A, B, and D are incorrect and are the result of common calculation errors or of incorrectly combining like and unlike terms.

The value 108 in the equation is the value of P in P = 108 − 23 d when d = 0. When d = 0, Kathy has worked 0 days that week. In other words, 108 is the number of phones left before Kathy has started work for the week. Therefore, the meaning of the value 108 in the equation is that Kathy starts each week with 108 phones to fix because she has worked 0 days and has 108 phones left to fix.

Choice A is incorrect because Kathy will complete the repairs when P = 0. Since P = 108 − 23d, this will occur when 0 = 108 − 23d or when d = \(\frac{108}{23}\), not when d = 108. Therefore, the value 108 in the equation does not represent the number of days it will take Kathy to complete the repairs. Choices C and D are incorrect because the number 23 in P = 108 − 23P = 108 indicates that the number of phones left will decrease by 23 for each increase in the value of d by 1; in other words, that Kathy is repairing phones at a rate of 23 per day, not 108 per hour (choice C) or 108 per day (choice D).

The total number of messages sent by Armand is the 5 hours he spent texting multiplied by his rate of texting: m texts/hour × 5 hours = 5m texts. Similarly, the total number of messages sent by Tyrone is the 4 hours he spent texting multiplied by his rate of texting: p texts/hour × 4 hours = 4p texts. The total number of messages sent by Armand and Tyrone is the sum of the total number of messages sent by Armand and the total number of messages sent by Tyrone: 5m + 4p.

Choice A is incorrect and arises from adding the coefficients and multiplying the variables of 5m and 4p. Choice B is incorrect and is the result of multiplying 5m and 4p. The total number of messages sent by Armand and Tyrone should be the sum of 5m and 4p, not the product of these terms. Choice D is incorrect because it multiplies Armand’s number of hours spent texting by Tyrone’s rate of texting, and vice versa. This mix-up results in an expression that does not equal the total number of messages sent by Armand and Tyrone.

To calculate (7 + 3i) + (−8 + 9i), add the real parts of each complex number, 7 + (−8) = −1, and then add the imaginary parts, 3i + 9i = 12i. The result is −1 + 12i.

Choices B, C, and D are incorrect and likely result from common errors that arise when adding complex numbers. For example, choice B is the result of adding 3i and −9i, and choice C is the result of adding 7 and 8.

Since k = 3, one can substitute 3 for k in the equation \(\frac{x − 1}{3}\) = k, which gives \(\frac{x − 1}{3}\) = 3. Multiplying both sides of \(\frac{x − 1}{3}\) = 3 by 3 gives x − 1 = 9 and then adding 1 to both sides of x − 1 = 9 gives x = 10.

Choices A, B, and C are incorrect because the result of subtracting 1 from the value and dividing by 3 is not the given value of k, which is 3.