Math 370 Exam 1 Review Name. Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.

Transcription

1 Math 370 Exam 1 Review Name Determine whether the relation is a function. 1) {(-6, 6), (-6, -6), (1, 3), (3, -8), (8, -6)} Not a function The x-value -6 corresponds to two different y-values, so this relation does not satisfy the definition of a function. Determine whether the equation defines y as a function of x. 2) x = y 2 y is not a function of x 3) x + y 3 = 27 y is a function of x Each value of x will correspond to exactly one value of y. Solve the problem. 4) The total cost in dollars for a certain company to produce x empty jars to be used by a jelly producer is given by the function C(x) = 0.7x + 29,000. Find C(70,000), the cost of producing 70,000 jars. $78,000 Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x. 5) function

2 6) not a function. This graph fails the Vertical Line Test. Use the graph to find the indicated function value. 7) y = f(x). Find f(-2) 3.6 Identify the intervals where the function is changing as requested. 8) Increasing (0, 5) and (11, 12)

3 The graph of a function f is given. Use the graph to answer the question. 9) Find the numbers, if any, at which f has a relative minimum. What are the relative minima? f has a relative minimum at x = -1 and 1; the relative minimum is 0 Determine whether the given function is even, odd, or neither. 10) f(x) = x 3-3x Odd 11) f(x) = x5 - x 4 Neither Based on the graph, find the range of y = f(x). 4 if -5 x < -3 12) f(x) = x if -3 x < 5 3 x if 5 x 12 [0, 5)

4 Solve the problem. 13) A gas company has the following rate schedule for natural gas usage in single-family residences: Monthly service charge $8.80 Per therm service charge 1st 25 therms Over 25 therms $0.6686/therm $ /therm What is the charge for using 25 therms in one month? What is the charge for using 45 therms in one month? Construct a function that gives the monthly charge C for x therms of gas. The charge for 25 therms is $ The charge for 45 thems is $42.69 C(x) = x if 0 x x if x > 25 The monthly charge C(x) can be expressed in a piecewise fashion, since the rate is less for the first 25 therms than it is for therms in excess of 25. For 0 x 25, the monthly charge will be x Thus we know that the first 25 therms will cost $ $0.6686(25) = $ If x > 25, then the charges will include the $ for the first 25 therms, along with the $ /therm for the therms used above 25 (that is, for the last x-25 therms). Thus, for x > 25, C(x) = (x-25) = x = x Find and simplify the difference quotient 14) f(x) = 5x 2 5(2x+h) f(x + h) - f(x), h 0 for the given function. h Use the given conditions to write an equation for the line in slope-intercept form. 15) Passing through (2, 5) and (4, 2) y = x + 8

5 Determine the slope and the y-intercept of the graph of the equation. 16) 2x - 6y - 12 = 0 m = 1 ; (0, -2) 3 Solve the given equation for y. This will be slope-intercept form, which makes the slope and y-intercept easily seen. Solve. 17) The average value of a certain type of automobile was $13,320 in 1995 and depreciated to $4020 in Let y be the average value of the automobile in the year x, where x = 0 represents Write a linear equation that models the value of the automobile in terms of the year x. y = -2325x + 13,320 Use the given conditions to write an equation for the line in the indicated form. 18) Passing through (4, 4) and perpendicular to the line whose equation is y = 1 3 x + 8; slope-intercept form y = - 3x + 16 Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. 19) h(x) = -(x + 7) 2 + 2

6 Begin by graphing the standard square root function f(x) = function. 20) h(x) = -x x. Then use transformations of this graph to graph the given Notice that the graph of the function g(x) = x would be the graph of y = x shifted 1 unit left and 2 units up. Replacing x with -x would then cause the graph of y = g(x) to be reflected about the y-axis. We might also notice that the domain of h(x) is restricted by the requirement that the radicand be nonnegative. -x x -1 x 1 So we could make a table of x and y values starting at x =1 and moving to the left.

7 Begin by graphing the standard absolute value function f(x) = x. Then use transformations of this graph to graph the given function. 21) g(x) = 1 4 x

8 Begin by graphing the standard cubic function f(x) = x 3. Then use transformations of this graph to graph the given function. 22) g(x) = -(x - 5) 3-2

9 Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g. 23) g(x) = -f(x - 2) - 2 y = f(x) Find the domain of the composite function f g. 24) f(x) = 6 x + 7, g(x) = x + 2 (-, -9) (-9, ) Find the inverse of the one-to-one function. 25) f(x) = (x + 3) 3 f -1 (x) = 3 x - 3 Recall that a function and its inverse "swap" x- and y-coordinates. So start with the equation y = (x + 3) 3 and interchange x and y. Then solve for y (which of course represents f -1 after the swap).

10 Use the graph of f to draw the graph of its inverse function. 26)

11 Graph f as a solid line and f -1 as a dashed line in the same rectangular coordinate space. Use interval notation to give the domain and range of f and f ) f(x) = x 2-6, x 0 f domain = [0, ); range = [-6, ) f -1 domain = [-6, ); range = [0, ) Find the distance between the pair of points. 28) (4, 5) and (-5, -6) 202

12 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Complete the square and write the equation in standard form. Then give the center and radius of the circle. 29) x 2 + y x + 8y = 8 A) (x + 5) 2 + (y + 4) 2 = 49 (-5, -4), r = 7 C) (x + 4) 2 + (y + 5) 2 = 49 (4, 5), r = 49 A B) (x + 4) 2 + (y + 5) 2 = 49 (-4, -5), r = 7 D) (x + 5) 2 + (y + 4) 2 = 49 (5, 4), r = 49 A) Complete the square in both x and y to put the equation in the desired form. B) C) D) Solve the problem. 30) The area of a rectangular garden is 225 square feet. The garden is to be enclosed by a stone wall costing $24 per linear foot. The interior wall is to be constructed with brick costing $9 per linear foot. Express the cost C, to enclose the garden and add the interior wall as a function of x. C(x) = 9x x x Divide and express the result in standard form. 31) 6 + 3i 7-5i i Perform the indicated operations and write the result in standard form. 32) ( )( ) 21 Start by rewriting the expression as ( 5-4i)( 5 + 4i). Solve the problem. 33) The profit that the vendor makes per day by selling x pretzels is given by the function P(x) = x x Find the number of pretzels that must be sold to maximize profit. 350 pretzels

13 Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 34) f(x) = -4x 4 + 3x 3 + 4x 2 + 3x + 5 falls to the left and falls to the right 35) f(x) = x - 2x 2-2x 3 rises to the left and falls to the right MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. 36) f(x) = x (x ) 4 A) B) - 1 4, multiplicity 2, touches the x-axis and turns around., multiplicity 2, crosses the x-axis. C) 1 4, multiplicity 2, touches the x-axis and turns around; 2, multiplicity 5, crosses the x-axis D) - 1 4, multiplicity 2, touches the x-axis and turns around; -2, multiplicity 5, crosses the x-axis A A) B) C) D)

14 Graph the polynomial function. 37) f(x) = x 4-4x 2

15 38) f(x) = x 3 + 2x 2-5x - 6 Solve the problem. 39) Solve the equation 3x 3-28x x - 20 = 0 given that 5 is a zero of f(x) = 3x 3-28x x , 4, 1 3 Find a rational zero of the polynomial function and use it to find all the zeros of the function. 40) f(x) = 2x 4-17x x 2-83x + 39 {1, 3, 3 + 2i, 3-2i} 2 Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. 41) x 4 + 2x 3-10x 2-14x - 3 = 0 {-1, 3, , -2-3}

16 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function. 42) f(x) = 6x 8-9x 7 + x 6-3x + 18 A) 4 or 2 positive zeros, no negative zeros B) 4 positive zeros, no negative zeros C) 4, 2 or 0 positive zeros, 1 negative zeros D) 4, 2 or 0 positive zeros, no negative zeros D A) B) C) D) Find the domain of the rational function. 43) h(x) = x + 7 x x {x x 0, x -36} Use the graph of the rational function shown to complete the statement. 44) As x 3 +, f(x)? - Find the vertical asymptotes, if any, of the graph of the rational function. 45) h(x) = x + 1 x 2-1 x = 1 Find the horizontal asymptote, if any, of the graph of the rational function. 46) g(x) = 10x2 2x y = 5

17 Use transformations of f(x) = 1 x or f(x) = 1 to graph the rational function. x2 47) f(x) = 1 x

18 1 48) f(x) = (x + 2) 2 + 4

19 Graph the rational function. 49) f(x) = 4x2 x 2-1

20 50) f(x) = x2-2x (x - 5) 2 Find the slant asymptote, if any, of the graph of the rational function. 51) h(x) = x3-3 x 2 + 6x y = x - 6

21 Graph the function. 52) f(x) = x2 + 4x - 6 x - 6 Solve the problem. 53) A company that produces inflatable rafts has costs given by the function C(x) = 25x + 20,000, where x is the number of inflatable rafts manufactured and C(x) is measured in dollars. The average cost to manufacture each inflatable raft is given by what function? _ C (x) = What is the horizontal asymptote for the function _ C? Describe what this means in practical terms. y = 25; $25 is the least possible cost for producing each inflatable raft.

22 Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. 54) (x - 4)(x + 3) > 0 (-, -3) (4, ) 55) x 2-4x [-2, 6] 56) (x + 5)(x + 1)(x - 2) > 0 (-5, -1) (2, ) 57) x 3 + 5x 2 - x - 5 > 0 (- 5, -1) (1, )

23 Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x - 1)(3 - x) 58) (x - 2) 2 0 (-, 1] [3, ) 59) 4x x + 7 < x (-7, -3) (0, )