precalc unit 1 and unit 2 study guide

In 2016, the cost to mail a package was $2.54 for up to ounces, plus an additional cost of $0.20 for each
additional ounce or portion of an ounce less than a full ounce. A portion of the graph of this relationship is given with cost, in dollars, as a function of ounces. Which of the following describes the restrictions on the range for such
a function?

In 2016, the cost to mail a package was $2.54 for up to ounces, plus an additional cost of $0.20 for each
additional ounce or portion of an ounce less than a full ounce. A portion of the graph of this relationship is given with cost, in dollars, as a function of ounces. Which of the following describes the restrictions on the range for such
a function?

D.) The range is values of the form 2.54 + 0.2x, where x is a nonnegative integer.

The table describes rates of change of a function f for selected intervals of x. The function f is defined for 0 ≤ x ≤ 4. On which of the following intervals is the graph of f concave down.

c.) 2 < x < 3

The graph of the function f is given for -3 ≤ x ≤ 6. Which of the following statements about the rate of change of f over the interval 2 < x < 6 is true?

d.) The rate of change is decreasing

A ball is thrown through an open window to the ground below. The height of the ball, in meters, at time t seconds after it is thrown can be modeled by the function h, given by h(t) = -4.9t^2 +4.4t + 15.24. Which of the following describes the height of the ball above the ground?

c.) After leaving the window, the height of the ball increases to its maximum height of 16.228 meters. Then the height of the ball decreases until it reaches the ground 1.820 seconds after reaching its maximum height.

The table gives characteristics of the rates of change of the function f on different intervals. Which of the following is true about f on the interval 3 < x < 4? (positive and decreasing)

a.) f is increasing, and the graph of f is concave down.

The polynomial function p is given by p(x) = (x+2)^4. Which of the following expressions is equivalent to (x+2)^4?

c.) x^4 + 8x^3 + 24x^2 + 32x + 16

The function f is given by f(x) = (x+3)^4. When f is rewritten in the form f(x) = x^4 + ax^3 + bx^2 + cx + d, which of the following values is greatest?

c.) c

The functions g and f are given by g(x) = 3x^2 - 2x and f(x) = 6x^4 + 5x^3 + 3x - 5. Which of the following statements is true about the remainder when f(x) is divided by g(x)?

d.) The remainder is (7x -5), so g(x) is not a factor of f(x), and the graph of y = f(x)/g(x) does not have a slant asymptote.

The figure shows the graph of a function f, the zero and extrema for f are labeled, and the point of inflection of the graph of f is labeled. Let A, B, C, D, and E represent the x-coordinates at those points. Of the following, on which interval is f increasing and the graph of f concave down?

a.) the interval from A to B

The depth of water, in feet, at a certain place in a lake is modeled by a function W. The graph of y = W(t) is shown for 0 ≤ t ≤ 30, where t is the number of days since the first day of a month. What are all intervals of t on which the depth of water is increasing at a decreasing rate?

a.) (3,6) only

A music agent is planning a series of concerts at local farms around the nation. The agent is building a model to estimate crowd capacity based on different sizes of square fields. Which type of function is most likely to model crowd capacity in this situation?

c.) quadratic

The table gives the average rates of change for the function f, g, h, and k for certain intervals of x. Which of the functions is best modeled by a piecewise-linear segments with different slopes?

c.) h

At a bakery, the number of cookies baked each day changes based on anticipated demand. The scatterplot shows the change in hundreds of cookies baked from the previous day for eight days. The point at (2,5) means that on day 2,
the number of cookies baked will be 500 more than the number of cookies baked on day 1. A function model C is
to be constructed for the number of cookies baked on each of the days, 0 through 8. Which of the following
statements best supports the selection of a model for C?

b.) Because the information about rate of change is roughly linear, a quadratic model is best for C

The ancient Pythagoreans studied figurate numbers, which are numbers that can be shown by taking dots or spheres and arranging them into geometric shapes. For example, the square numbers are 1, 4, 9, 16, 25, etc., and each of these numbers of dots can be arranged into a square. The tetrahedral numbers similarly specify the number of can be arranged into a square. The tetrahedral numbers similarly specify the number of
spheres needed to create a tetrahedron, which is a triangular-based pyramid The tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, etc. Which of the following statements is true?

d.) The tetrahedral numbers are best modeled by a cubic function because the 3rd differences are a nonzero constant.

The figure shown is the graph of a polynomial function g. Which of the following could be an expression for g(x)?

c.) 0.25(x-5)^2(x-1)(x+8)

The polynomial function p is an odd function. If p(3) = -4 is a relative minimum of p, which of the following statements about p (-3) must be true?

c.) p(-3) = 4 is a relative minimum.

The function f is given by f(x) = 3x^2 + 2x + 1. The graph of which of the following functions is the image of the graph of after a vertical dilation of the graph of f by a factor of 2?

b.) k(x) = 6x^2 + 4x + 2, because this is a multiplicative transformation of f that results from multiplying f(x) by 2

The polynomial function p is given by p(x) = (x+3)(x^2 - 2x - 15). Which of the following describes the zeros of p?

a.) p had exactly two distinct real zeros

A polynomial function p is given by p(x) = -x(x-4)(x+2). What are all the intervals on which p(x) ≥ 0?

(-∞, -2] U [0,4]

The table gives values of a polynomial function Q for selected values of x. What is the degree of Q?

c.) 4

The leading term of the polynomial function p is asubnx^n, where asubn is real number and n is positive integer. The factors p include (x-3), (x-i), and (x-(2+i)). What is the least possible value of n?

c.) 5

The function Q is a polynomial of degree 3. If Q(5) = 0, which of the following must be true?

c.) Q(x) can be expressed as (x-5)(P(x)), where P(x) is a polynomial of degree 2.

For the polynomial function g, the rate of change of g is increasing for x < 2 and decreasing for x > 2. Which of the following must be true?

d.) The graph of g has point of inflection at x = 2, is concave up for x < 2, and is concave down for x > 2.

The polynomial function f is given by f(x) = (x-4)(3x-1)^2. Which of the following descriptions of f is true?

d.) f is a polynomial of degree 3 with a leading coefficient of 9.

A polynomial function has the form p(x) = ax^j + bx^k, where a and b are nonzero constants, j and k are nonnegative integers. Which of the following conditions guarantees that p is an even function?

b.) j = 2k and k is even

Values of the polynomial function f for selected values of x are given in the table. If all of the zeros of the function f are given in the table, which of the following must be true?

b.) The function f has a local minimum at (5,0)

The graph of the polynomial function g is shown. The function f is defined for 0 ≤ x ≤ 3 and is identical to the function g on that interval. How many total local minima and local maxima does the function f have?

b.) four

The polynomial function f is given by f(x) = ax^4 + bx^3 + cx^2 + dx + k, where a ≠ 0 and b, c, d, and k are constants. Which of the following statements about f is true?

b.) f has either a global maximum or a global minimum, but not both

The graph of the polynomial function f is shown. How many points of inflection does the graph of f have one the given portion of the graph?

b.) three

The table gives values of the function f for selected values of x. If the function f is linear, what is the value of f (13)?

d.) 34/3

Two drones are flying over a given area, and their heights above the ground are changing. The table gives the change in height, in feet, for the drones over successive 6-second intervals. Which of the following is true about the average rates of change for drone A and drone B over the time interval from t = 0 seconds to t = 30 seconds?

a.) The average rates of change are equal.

The table gives the average of rates of change of a function f over different intervals. On which of the intervals does the function increase the most?

d.) 8 ≤ x ≤ 10

The function f is defined for all real values of x. For a constant a, the average rate of change of f from x = a to x = a +1 is given by the expression 2a + 1. Which of the following statements is true?

d.) The average rate of change of f over consecutive equal-length input-value intervals is increasing at a constant rate, so the graph of f could be a parabola that opens up.

The graph of the function y = g(x) is given. Of the following, on which interval is the average rate of change of g least?

b.) -1 ≤ x ≤ 0

The average rates of change of the quadratic function p is -4 on the interval 0 ≤ x ≤ 2 and -1 on the interval 2 ≤ x ≤ 4. What is the average rate of change of p on the interval 6 ≤ x ≤ 8?

c.) 5

The table gives values of a function f for selected values of x. Which of the following conclusions with reason is consistent with the data in the table?

d.) f could be a quadratic function because the rates of change over consecutive equal-length intervals in the table can be described by y = 2x +1.

The daily high temperature at a certain point in a river is modeled by the graph. Each point on a vertical gridline indicates the temperature, in degrees Celsius, on the first day of the indicated month. Of the following, on the first day of which month is the rate of change of the temperature the greatest?

b.) may

The function f is not explicitly given. The function g is given by g(x) = f(x+1) - f(x). The function h is given by h(x) = g(x+1) - g(x). If h(x) = -6 for all values of x, which of the following statements must be true?

d.) Because h is negative and constant, g is decreasing, and the graph of f is concave down.

The function has a negative average rate of change on every interval of in the interval 0 ≤ x ≤ 10. The function g has a negative average rate of change on every interval of x in the interval 5 < x ≤ 10. Which of the following statements must be true about the function h, defined by h(x) = f(x) + g(x), on the interval 0 ≤ x ≤ 10?

d.) h is decreasing on 0 ≤ x < 4; h can be increasing, decreasing, or both increasing and decreasing on 5 < x ≤ 10.

The rational function h is expressed as the quotient of two polynomial functions f and g by h(x) = f(x)/g(x). The function f is given by f(x) = 6x^3 + x^2 + 60x - 25. If the graph of has a slant asymptote of y = 2x - 1, which of the following describes g?

a.) g has degree 2 with leading coefficient 3

A polynomial function has three distinct zeros each with multiplicity 1, and its leading coefficient is positive. The polynomial function has exactly one zero with multiplicity 3, and its leading coefficient is negative. The rational function h can be written as the quotient of p and q by h(x) = P(x)/q(x). Which of the following statements about h must be true?

c.) The graph of h has a horizontal asymptote at y = a, where a < 0.

The function g is given by g(x) = x^3 - 3x^2 - 18x, and the function h is given by h(x) = x^2 - 2x - 35. Let k be the function given by k(x) = h(x)/g(x). What is the domain of k?

c.) all real numbers x where x ≠ -3, x ≠ 0, x ≠ 6

Which of the following functions has a zero at x = 3 and has a graph in the xy-plane with a vertical asymptote at x = 2 and a hole at x = 1?

a.) h(x) = (x^2-4x+3)/(x^2-3x+2)

In the xy-plane, the graph of a rational function f has a vertical asymptote at x = -5. Which of the following expressions could define f(x)?

d.) ((x-5)(x-3))/((x-3)(x+5))

In the xy-plane, the graph of the rational function f has a vertical asymptote at x = 4. Which of the following expressions could define f(x)

a.) ((x+1)^3(x-4)^2)/((x+1)^2 (4-x)^3)

The graphs of the polynomial functions f and g are shown. The function h is defined by h(x) = f(x)/g(x). What are all vertical asymptotes of the graph of y = h(x)?

c.) x = -2 and x = 3 only

Let f be a rational functional that is graphed in the xy-plane. Consider x = 1 and x = 7. The polynomial in the numerator of f has a zero at x = 1 and does not have a zero at x = 7. The polynomial in the denominator of f has zeros at both x = 1 and x = 7. The multiplicities of the zeros at x = 1 in the numerator and in the denominator are equal. Which of the following statements is true?

c.) The graph of f has a hole at x = 1 and a vertical asymptote x = 7.

The rational function g is given by g(x) = ((x^2+3x)(x^2-4x-5))/((x+3)(x-1)(x-2)). For what input values of g are the output values of g equal to 0?

b.) -1, 0, and 5 only

The rational function r is given by r(x) = (x^3 +4x^2 +4x)/(x^2-9). On what intervals of x is r(x) ≥ 0?

d.) -3 < x ≤ 0 and x > 3

Which of the following names a function with a hole in its graph at x = 1 and provides correct reasoning related to the hole?

a.) The graph of f(x) = (x^2 - 1)/(x-1) has a hole at (1,2) because the values of (x^2 - 1)/(x-1) get arbitrarily close to 1, but the function is undefined at x = 1.

The rational function g is given by g(x) = (x^3+1000)/(x^2-100) = ((x+10)(x^2-10z+100))/(x^2-100). Which of the following statements describes the behavior of the graph of g?

b.) The graph has hole at x = -10 because (x+10) appears exactly once, when both the numerator and the denominator of g are factored.

The functions and are defined for all real numbers such that g(x) = f(2(x-4)). Which of the following sequences of transformations maps the graph of f to the graph of g in the same xy-plane?

d.) A horizontal dilation of the graph of f by a factor of 1/2, following by a horizontal translation of the graph of f by 4 units.

The function f is given by f(x) = x^2 +2. the function g is the result of a transformation of f and is given by g(x) = x^2/4 + 2. Which of the following describes the transformation of the graph of f whose image is the graph of g?

c.) a horizontal dilation by a factor of 2

The function f has domain [-2,2] and range [1,5]. The function g is given g(x) = -2f(x+3) + 4. What are the domain and range of g?

a.) domain: [-5,-1], range: [-6,2]

The speed of a car traveling on a highway is being recorded once per second for two minutes. During this time interval, the car gradually speeds up slightly to pass another vehicle, then the car returns to its original speed. The recorded speed of the car with respect to time can be modeled by linear, quadratic, and exponential functions. For
each of the three models, their residuals are small and are without pattern. Which of the following conclusions is best?

b.) A quadratic model is best based on contextual clues

The functions f and g are given by f(x) = x^2 + 1 and g(x) = 4x - 1. Which of the following is an expression for f(g(x))?

c.) 16x^2 - 8x + 2

The functions f and g are given by f(x) = 1/x and g(x) = sqrt(x). What is the domain of the function h given by h(x) = f(g(x))

c.) all real numbers greater than 0

The function p (not shown) is a polynomial function of degree 3. The graphs of four functions f, g, h, and k are given. The output values of p are the same as the output values of the composition function when p is composed with one of these functions as the input function. For which of the functions is this statement true?

a.) f

The function f is given by f(x) = 9(25)^x. Which of the following is an equivalent form for f(x)?

d.) f(x) = 9(5)^2x

The value, in millions of dollars, of transactions processed by an online payment platform is modeled by the
function M. The value is expected to increase by 6.1% each quarter of a year. At time t = 0 years, 54 million dollars of transactions were processed. If t is measured in years, which of the following is an expression for M(t)?

d.) 54(1.061)^4t

Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after d days can be modeled by the function h given by h(d) = Asub0(0.5)^(d/8), where Asub0 is the amount of iodine-131 in the sample at time d=0. Which of the following functions k models the amount of iodine-131 remaining after t hours, where Asub0 is the amount of iodine-131 in the sample at time t = 0?

k(t) = Asub0(0.5^(1/192))^t

The exponential function f is defined by f(x) = ab^x, where a and b are positive constants. The table gives values of f(x) at selected values of x. Which of the following statements is true?

d.) f demonstrates exponential growth because a > 0 and b > 1.

The function g is a function of the form g(x) = ab^x where a ≠ 0 and b > 0. The function f is given by f(x) = g(x) + 4. Which of the following statements is true?

c.) The output values of g only, not f, are proportional over equal-length input-value intervals.

The function m is given by m(x) = 36^(x/2). Which of the following expressions could also define m(x)?

6^x

The function h is given by h(x) = 5(3)^(-x/2). What is the value of h(1)?

d.) 5/sqrt(3)

The function f is given by f(x) = 2^x, and the function g is given by g(x) = f(x)/8. For which of the following transformations is the graph of g the image of the graph of f?

b.) A horizontal translation to the right 3 units

The function f is given by f(x) = 3^x. The function g is given by g(x) = (f(x))^b, where b < 0. Which of the following describes the relationship between the graphs of f and g?

b.) The graph of g is a combination of a horizontal dilation of the graph of f and a reflection over the y-axis.

The functions f and g are given by f(x) = 2^x and g(x) = (2^x)(2^a), where a > 0. Which of the following describes the relationship between the graph of f and the graph of g?

d.) The graph of g is a horizontal translation of the graph of f by -a units.

Water hyacinth is an invasive plant species found in many lakes that typically grows at a rate of 7% per day. As part of a study, a scientist introduces a 150-gram sample of water hyacinth into a testing pool. Which of the following functions gives the amount of water hyacinth in the testing pool t weeks after the sample is introduced?

d.) k(t) = 150(1.07^7)^t

The table gives values for the functions and at selected values of x. Functions f and g are defined for all real numbers. Let h be the function defined by h(x) = f(g(x)). What is the value of h(0)?

a.) -2

The graph of he piecewise-linear function f is shown in the figure. Let g be the inverse function of f. What is the maximum value of g?

c.) 5

The function f is defined by f(x) = sqrt(4-x^2) for -2 ≤ x ≤ 0. Which of the following expressions defines f^-1(x)?

c.) -sqrt(4-x^2) for 0 ≤ x ≤ 2

The function g is given by g(x) = (4x+6)/5. Which of the following defines g^-1(x)?

d.) (5x-6)/4

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

c.) f^-1(x) = sqrt((x-3)/4) for x ≥ 3

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?

d.) W^-1 is a decreasing function of the amount of water in the tank

If m = logsub3(81), which of the following is also true?

b.) 3^m = 81

The sales of a new product, in items per month, is modeled by the expression 225 + 500logsub10 (15t + 10), where t represents the time since the product became available for purchase in months. What is the number of items sold per month for time = 6?

b.) 1225

Consecutive terms of a sequence have the values 2, -1, 1/2, -1/4, and 1/8. Of the following, which describes the sequence?

b.) The terms could be part of a geometric sequence with a common ratio of -1/2.

The terms of the increasing arithmetic sequence asubn are positive. The terms of the increasing geometric sequence gsubn are positive. The values of the first terms of both sequences are the same, and the values of the fourth terms of both
sequences are the same. Which of the following statements describes the values of the second terms of the
sequences?

b.) The second term of the arithmetic sequence must be greater than the second term of the geometric sequence

The fifth term of a geometric sequence is 24, and the sixth term is 48. What is the value of the tenth term?

c.) 768

The first term of an arithmetic sequence is 5, and the common difference of the sequence is 2. What is the eighth term of the sequence?

a.) 19

Consecutive terms of a sequence have the values of 6, 2, -2, and -6. Of the following, which describes the sequence?

a.) The terms could be part of an arithmetic sequence with a common difference of -4

The general term of a sequence is given by asubn = 51 + 3 (n-10), where asub0 is the initial value. Which of the following expressions also gives the general term of the sequence?

c.) 21 + 3n

For an arithmetic sequence Ssubn, Ssub3 = 3 and Ssub6 = 24. What is the value of Ssub10 - Ssub8

c.) 14

A family needs to buy one shovel and between one and eight plants, inclusive, for their new garden. The cost of the shovel is s dollars, and the cost of one plant is p dollars. The output values of which of the following give the possible costs for these items, in dollars?

c.) The arithmetic sequence Csubn = s + pn for 1 ≤ n ≤ 8

Which of the following includes the input-output pairs (2,4) and (3,8)?

d.) The exponential function h(n) = 2(2)^(n-1)

An exponential function G has a known common ratio of 1/2 and includes the input-output pair (1,4). Which of the following could define G(x)?

d.) 4(1/2)^(x-1)

The second term of a sequence is 6, and the fourth term is 24. Of the following, which statement is true?

c.) If the sequence is geometric, the fifth term could be 48.

The functions f and g are given by f(x) = 4^(5x-1) and g(x) = 8^(x/4). When solving the equation f(x) = g(x), the functions can be rewrittten in equivalent forms so that the equation can be solved without the use of technology. Which of the following are equivalent definitions of f and g that aid in solving f(x) = g(x) without the use of technology?

a.) f(x) = 2^(logsub2(4)(5x-1)) and g(x) = 2^(logsub2(8)*x/4))

The function f is given by f(x) = 4(2)^(x-3). If the function g is the inverse of f, which of the following could define g(x)?

c.) logsub2(x/4) + 3

The function f is given by f(x) = logsub2(logsub3(x)). Which of the following is an expression for f^-1(x)?

b.) 3^2^x

The table gives values of the function f for selected values of x. Which of the following is a verbal representation of f^-1(x), the inverse function of f?

b.) f^-1(x) is logarithmic with output values increasing by 1 every time input values double

The exponential function g is given by g(x) = 5^x. Which of the following expressions defines g^-1(x)?

a.) logsub5(x)

The function f is an increasing function such that every time the output values of the function f increase by 1, the corresponding input values multiply by 4. Which of the following could define f(x)?

d.) logsub4(x)

Let k, w, and z be positive constants. Which of the following is equivalent to logsub10(kz/w^2)?

b.) logsub10(k) + logsub10(z) - 2logsub10(w)

Let x and y be positive constants. Which of the following is equivalent to 2lnx-3lny?

a.) ln(x^2/y^3)

What are all the values of x for which ln(x^3) - lnx = 4?

c.) x = e^2 only

The range of function f is the positive real numbers. The function g is given by g(x) = ln(f(x)). Solutions to which of the following equations are useful in solving g(x) = 2?

b.) f(x) = e^2

To solve the equation logsub8(x-3) + logsub8(x+4) = 1, one method is to apply the properties of logarithms to rewrite the equation in an equivalent form. This equivalent equation can be used to identify possible solutions. Of the following, which is such an equation?

d.) x^2 + x - 12 = 8

Which of the following is the inverse of the function f given by f(x) = 4logsub2(x=3) - 1?

d.) g(x) = 2^((x+1)/4) - 3