2A AP Precalc Review (2.1-2.8) Review

​In the xy-plane, the graphs of the linear function L and the exponential function E both pass through the points (0,2) and (1,6). The function f is given by fx=Lx-Ex. What is the maximum value of f ?

0.540 Using the two points and the form of y=b+mx, it can be determined that the initial value b is 2 and the constant rate of change (slope) m is 4. Therefore, Lx=2+4x. Using the two points and the form of y=abx, it can be determined that the initial value a is 2 and the constant proportion (common ratio) b is 3. Therefore, Ex=2·3x. The maximum value of fx=2+4x-2·3x can be found using the graphing calculator and occurs at the point (0.545,0.540)

The function f is given by f(x)=1000^x. If f(x)=10, what is the value of x ?

1/3 Because 10^3=1000, the cube root of 1000 is 10 and can be written as cube root 1000=1000^1/3.

The function f is given by f(x)=x+2. If f^-1(n)=4, what is the value of n ?

6 The inverse function f-1 maps the output values of f to the input values of f. Because the output value of f is n and the input value of f is 4, the solution is found by evaluating f4.

The first term of a geometric sequence is 2/3. The next three terms are 1/2, 3/8, and 9/32. What is the tenth term of the sequence?

6561/131,027 The tenth term is the first term times the common ratio 9 times or 2/3(3/4)^9.

The function f is given by fx=k+a·bx, where a<0, b>1, and k is a real number. Which of the following is true? Responses A ​limx→-∞fx=-k B ​limx→-∞fx=k C ​limx→∞fx=-k D ​limx→∞fx=k​

Answer B Correct. As x decreases without bound, the values of a·bx increase from large magnitude negative values toward 0. As a result, the values of f get closer to k.

How do you know if a model is appropriate based off a residual plot.

The residuals appear without pattern. Therefore, the model is appropriate.

The number of bacteria in a dish is 1150 at time t=0 hours. The number of bacteria in the dish doubles every 3 hours. If the function B gives the number of bacteria in the dish at time t hours for t≥0, which of the following could define Bt ?

​1150·2^(t/3) Because the doubling of the bacteria occurs every 3 hours, the initial value should be multiplied by 2 only once for each multiple of 3.

The function f is defined by fx=(4x^2)+3 for x≥0. Which of the following expressions defines the inverse function of f ?

​f-1x=(sqrtx-3/4) for X>=3 This is the result of reversing the roles of x and y in the equation y=4x^2+3and then solving for y. Note that the values in the domain of f-1 correspond to the values in the range of f.

The function h is given by hx=a·b^(-x), where a<0 and b>1. Which of the following describes h and its graph in the xy-plane?

​h is increasing, and the graph of h is concave down. The negative exponent property states that b-x=1bx. As x increases without bound, 1bx gradually gets closer to 0, indicating exponential decay. An exponential decay function is decreasing, and its graph is concave up. However, the negative value of a reflects the graph of the exponential decay function over the x-axis, which results in the function being increasing and its graph concave down.

A water tank is leaking water from a crack in its base. The amount of water, in hundreds of gallons, remaining in the tank t hours after the crack formed can be modeled by W, a decreasing function of time t. Which of the following gives a verbal representation of the function W-1, the inverse function of W ?

​​W^-1 is a decreasing function of the amount of time after the crack formed. Because W maps times to amounts of water, the inverse function maps amounts of water to times. Because W is decreasing, as one variable increases, the other decreases. This relationship does not change for the inverse function.

A drop of water hits the surface of a lake and forms concentric circular ripples. The radius, in inches, of the circle enclosed by the outer ripple increases as a function of time t, in minutes, and is modeled by the function r given by rt=25sqrt2t+1. Which of the following gives the area of the circle enclosed by the outer ripple, in square inches, as a function of time t, in minutes? (Note: The area of a circle with radius x is given by A=πx^2.) Responses A ​50πt+25π B ​1250πt+625π C ​25sqrt2πt^2+1 D ​25πt^2sqrt2t+1

Answer B Correct. This is the result of determining the area function as a composition function Ax=πx^2, and writing as Art=πrt^2. This results in πsqrt252t+1^(2)=π(625)(2t+1)=1250πt+625π.

The function g is given by gx=4·2^(3+x.) The function g can also be expressed as which of the following? Responses A ​32+2^x B ​32·2^x C ​32+8^x D ​512·8^x

Answer B Correct. Using the product property for exponents, 4·2^(3+x)=4·2^3·2^x=32·2^x.

A certain hobby shop wants to attract customers on slow sales days by offering a new discount. The shop offers a 1.5% discount on Wednesdays with an added bonus for repeat customers. For customers who make purchases on successive Wednesdays, their discount is doubled from the previous Wednesday. For example, a certain customer makes purchases on three successive Wednesdays and receives 1.5% (week 1), 3% (week 2), and 6% (week 3) discounts on those three Wednesdays. Based on this pattern, what is the first successive week that a customer's discount would exceed the purchase price of an item? Responses A ​6 B ​7 C ​8 D ​11

Answer C Correct. The contextual scenario can be modeled by the exponential inequality 1.5·2^(x-1)≥100 and solved for x≥7.059. On the eighth successive week, the discount will be greater than 100% of the purchase price of an item.

​The function g is given by gx=4x+6/5. Which of the following defines g^-1(x) ? Responses A ​5/4x+6 B ​5x+6/4 C ​(5x/4) -6 D ​5x-6/4

Answer C ​Incorrect. This is the result of an error in the order of operations when reversing the operations of g to find the inverse function.

A ball is dropped from a height of 10 feet. The ball repeatedly bounces, and the maximum height of the ball after each bounce is 20% less than the previous maximum height. Which of the following arguments is correct regarding a sequence that can be used to model the successive maximum heights, in feet? A ​An arithmetic sequence is appropriate because each successive height would be found by subtracting 2 from the previous height. B ​An arithmetic sequence is appropriate because each successive height would be found by multiplying the previous height by 0.8. C ​​A geometric sequence is appropriate because each successive height would be found by subtracting 2 from the previous height. D ​​A geometric sequence is appropriate because each successive height would be found by multiplying the previous height by 0.8.

Answer D Correct. Subtracting 20% from the previous height is equivalent to multiplying the previous height by 0.8. In a geometric sequence, each successive term is found by multiplying the previous term by the common ratio.

The fourth term of a geometric sequence is 7, the fifth term is 28/3, and the kth term is 7168/243. The solution to which of the following equations gives the value of k ? A ​4(3/4)^x-7=7168/243 B ​7(3/4)^x-4=7168/243 C ​4(4/3)^x-7=7168/243 D 7(4/3)^x-4=7168/243

Answer D Correct. The correct common ratio is identified from the given terms by calculating the ratio of the fifth term to the fourth term. The x-value of the fourth term is appropriately indexed with the exponent. Therefore, the solution to the equation will give the value of k.

The first term in a geometric sequence is g1=5. For n≥2, the terms of the geometric sequence are given by gn=3·gn-1. For which of the following functions is gn=fn for positive integer values of n ? A ​fx=3·5^n B ​fx=3·5^n-1 C ​fx=5·3^n D ​5(3)^n-1

Answer D Correct. The function value corresponding to an index value of 1 is the initial value of the geometric sequence, g1=5. The common ratio of the function is the factor 3 by which the terms of the geometric sequence are successively multiplied.

Terms in a sequence are given by sn=4n+7, for n=0,1,2,3,.... Which of the following describes the sequence? A The sequence is geometric with an initial value of 7 and a common ratio of 4. B ​The sequence is geometric with an initial value of 11 and a common ratio of 4. C ​​The sequence is arithmetic with an initial value of 7 and a common difference of 4. D ​​​The sequence is arithmetic with an initial value of 11 and a common difference of 4.

C The initial value when n=0 is 7. Each time the value of n increases by 1, the terms in the sequence increase by the common difference (or constant rate of change) of 4.