Ch2 practice test. for the following functions. f (x) = 6x 2 + 2, Find the domain of the function using interval notation:

Transcription

1 Ch2 practice test Find for the following functions. f (x) = 6x 2 + 2, Find the domain of the function using interval notation: A hotel chain charges $75 each night for the first two nights and $55 for each additional night's stay. The total cost T is a function of the number of nights x that a guest stays. Find the expressions for a and b in the piecewise function defined above. In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum is 40. The fine F for violating these limits is $13 for every mile above the maximum or below the minimum. Find the expressions for a(x), b(x) and c(x) in the piecewise function defined above. Determine whether the curve is the graph of a function of x. Determine whether the equation defines y as a function of x. x 2 + (y - 9) 2 = 64

2 Determine whether the equation defines y as a function of x. If it does, state y as a function of x. Find a function whose graph is the given curve: The bottom half of the circle x 2 + y 2 = 81 The graph shown gives a salesman's distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day. Use the words at home, stationary, travelling away from home, or travelling toward home. (a) At 8:00 A.M. the salesman (b) From 8:00 A.M. until 9:00 A.M. the salesman is (c) From 9:00 A.M. until 10:00 A.M. the salesman is (d) From 10:00 A.M. until noon the salesman is (e) From noon until 1:00 P.M. the salesman is (f) From 1:00 P.M. until 3:00 P.M. the salesman is (g) From 3:00 P.M. until 5:00 P.M. the salesman is (h) From 5:00 P.M. until 6:00 P.M. the salesman is (i) From 6:00 P.M. until 7:00 P.M. the salesman is Westside Energy charges its electric customers a base rate of $6 per month, plus 13 per kilowatt-hour (kwh) for the first 300 kwh used and 7 per kwh for all usage over 300 kwh. Suppose a customer uses x kwh of electricity in one month. The function below gives the cost for electricity in dollars. Express the monthly cost E as a function of x. (ie, Find a and b. The graph of a function is given. Determine the interval(s) on which the function is increasing. The graph of a function is given.

3 Determine the interval(s) on which the function is decreasing. The graph of a function is given. Determine the average rate of change of the function between x = 1 and x = 3. The graph of a function is given. Determine the average rate of change of the function between the indicated values of the variable. A function is given. Determine the average rate of change of the function between the given values of the variable: f(x) = -5x 2 ; x = 5, x = 5 + h A function is given. Determine the average rate of change of the function between the given values of the variable: The graphs of f and g are given. Find a formula for the function f, if g(x) = 4x 3. The graphs of f and g are given. Find a formula for the function g, if f(x) = x.

4 The graph of y = f (x) is given. Match each equation with its graph. (a) y = f (x - 4) (b) y = f (x) + 3 (c) y = 2 f (x + 4) (d) y = -f (2x) The graph of y = f (x) is given. Match each equation with its graph. (a) y = -1/3 f (x) (b) y = -f (x + 3) (c) y = f (x - 3) + 4 (d) y = f (-x) A function f is given, and the indicated transformations are applied to its graph (in the given order) to obtain the function g. Write the equation for g. f(x) = x 5 ; shift left 6 units, stretch vertically by a factor of 3, and reflect about the y-axis reflect in the y-axis, shrink vertically by a factor of 1/2, and shift upward 3/7 unit f(x) = x ; shift to the left units, shrink vertically by a factor of 0.5, and shift down 6 units.

5 Sketch the graph of the function, not by plotting points, but by starting with the graph of y = x and applying transformations. Indicate below the steps you would take to graph the function: y 2 x 2 Determine whether the function f is even, odd, or neither. f(x) = x -2 f (x) = x -3 f(x) = x 6 + 4x 3 f (x) = x 4-5x 2 f(x) = x 11-6x 5 For the quadratic function y = 2x 2 + 6x, (a) Express the quadratic function in standard form. (b) Find the coordinates of the vertex, x-intercept(s), y-intercept. (c) Select the correct graph. For each given quadratic functions f (x) = 5x + x 2 f (x) = 3-16x - 16x 2 (a) Express the quadratic function in standard form. (b) Find its maximum or minimum value. Find a function whose graph is a parabola with vertex (3, 4) and that passes through the point (1, -12). Use the function whose graph is shown to answer the following questions. (a) Find the local maximum. (b) Find the local minimum. If a ball is thrown directly upward with a velocity of 49 ft/s, its height (in feet) after t seconds is given by y = 49t - 16t 2. What is the maximum height attained by the ball? A ball is thrown across a playing field. Its path is given by the equation below, where x is the distance the ball has traveled horizontally, and y is its height above ground level, both measured in feet. y = x 2 + x + 10 (a) What is the maximum height attained by the ball? (b) How far has it traveled horizontally when it hits the ground? Round your answer to the nearest foot. A soft-drink vendor at a popular beach analyzes his sales records, and finds that if he sells x cans of soda pop in one day, his profit (in dollars) is given by the following equation. P(x) = x 2 + 5x (a) What is his maximum profit per day? (b) How many cans must he sell for maximum profit?

6 The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If n trees are planted on an acre of land, then each tree produces n apples. So the number of apples produced per acre is given by: A(n) = n( n) How many trees should be planted per acre in order to obtain the maximum yield of apples? A rectangle has an area of 17 m 2. Find a function that models its perimeter P in terms of the length of one of its sides x. Find a function that models the surface area S of a cube in terms of its volume V. Find a function that models the area A of a circle in terms of its circumference C. A right triangle has one leg three times as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg. A wire L = 16 cm long is cut into two pieces, one of length x and the other of length (L - x), as shown in the figure. Each piece is bent into the shape of a square. (a) Find a function that models the total area A enclosed by the two squares in terms of x. (b) Find the value of x that minimizes the total area of the two squares. A graphing calculator is recommended. Find the dimensions that give the largest area for the rectangle shown in the figure. Its base is on the x-axis and its other two vertices are above the x-axis, lying on the parabola y = k - x 2, k = 10. (Give each answer correct to two decimal places.) Find the domain of the function. f ( x) x 10 x,, Use the given functions to evaluate the expressions below. f (x) = 3x - 5 g (x) = 4 - x 2 f (g (-5)) g (f (-5)) f (f (4)) g(g (3)) (f g)(x) (g f)(x).

7 Use the given graphs of f and g to evaluate the expression. f(g(-4)) g(f(-2)) (f g)(-4) (g g)(-4) Use the given functions to answer the following questions: (a) Evaluate (f g)(x) and find its domain. (b) Evaluate (g f)(x) and find its domain. Find the function f g h for f (x) = 1/ x, g(x) = x 3, and h(x) = x Consider the given graphs of functions. Determine which function is a one-to-one. Which of the given functions is a one-to-one function? Assume f and g are one-to-one functions. If f(x) = -x + 5, find f -1 (14)

8 If g(x) = x 2 + 4x with x -2, find g -1 (32). Find the inverse function of f where,, f( x) 7x 5, f (x) = 2 - x 3