Unit 2 Functions HW #1 Mrs. Dailey

Transcription

1 HW#1 Name Algebra II Unit 2 Functions HW #1 Mrs. Dailey 1) Determine for each of the following graphed relationships whether y is a function of x using the Vertical Line Test. 2) What are the outputs for an input of x= -5 given functions defined by the following formulas: (a) y = 50 - x 2 (b) y= 2 x

2 3) Evin is walking home from the museum. She starts 38 blocks from home and walks 2 blocks each minute. Evin s distance from home is a function of the number of minutes she has been walking. (a) Which variable is independent (input: x) and which variable is dependent (output: y) in this scenario? (b) Fill in the table below for a variety of time values. Time, t, in minutes Distance from home, D, in blocks (c) Determine an equation relating the distance, D, that Evin is from home as a function of the number of minutes, t, that she has been walking. (d) Determine the number of minutes, t, that it takes for Evin to reach home. (Distance = 0 miles from home) 4) In one of the following tables, the variable y is a function of the variable x. Explain which relationship is a function and why the other is not. Function: Only 1 y-value for every x-value.

3 HW#2 Name Algebra II Unit 2 Functions HW #2 Mrs. Dailey 1) Without using your calculator, evaluate each of the following given the function definitions and input values. (a) f(x) = 3x+7 (b) g(x) = 3x 2 (c) h( x) x 5 f(2) = g(-3)= h(14)= 2) Evaluate each of the following more complex functions. (a) 2 3x 5 f( x) 4x 10 (b) 2 25 x gx ( ) (c) h(x) = 30(1.2) x x f(-5) = g(4) = h(3)= 3) Ian invested $2500 in an investment vehicle that is guaranteed to earn 4% interest compounded yearly. The amount of money, A, in his account as a function of the number of years, t, since creating the account is given by the equation A(t) = 2500(1.04) t. (a) Evaluate A(0) and A(10). (b) What do the two values that you found in part (a) represent?

4 4) Based on the graph of the function y = g(x) shown below, answer the following questions. (a) Evaluate g(-2), g(0), g(3), and g(7). g(-2) Find y when x = -2 (b) What values of x solve the equation g(x) = 0? (c) Graph the horizontal line y = 2 on the grid above and label. (d) How many values of x solve the equation g(x) =2? Review 1) Find the product for each, a) (2x yz)(x+ yz) b) (x - 4) 2

5 HW#3 Name Algebra II Unit 2 Functions HW #3 Mrs. Dailey 1) Given f(x) = 3x 4 and g(x) = -2x+7 evaluate: (a) f(g(0)) g(0) = -2(0) + 7 g(0) = 7 f(7) = 3(7) 4 f(7) = 21-4 f(7) = 17 (b) f(f(3)) (c) (g ᵒ f)(6) Rewrite g(f(6) (d) (f ᵒ g)(5) 2) Given h(x) = x and g( x) x 2 evaluate: (a) h(g(18)) (b) (g ᵒ g)(11) (c) h(h(0)) (d) (g ᵒ h)(0)

6 3) The graphs of y = h(x) and y=k(x) are shown below. Evaluate the following based on these two graphs. (a) h(k(-2)) (b) (k ᵒ h)(0) (d) h(h(-2)) Find k(-2) -4 Find h(-4) -5 Answer: -5 4) If g(x) = 3x-5 and h(x) = 2x 4 then (g ᵒ h)(x)? (1) 6x 17 (2) 6x 14 (3) 5x -9 (4) x - 1 5) If f(x) = x and g(x) = x + 4 then f(g(x))? (1) x (2) x 2 +8x + 21 (3) 4x (4) x

7 6) Physics students are studying the effect of the temperature, T, on the speed of sound, S. They find that the speed of sound in meters per second is a function of the temperature in degrees Kelvin, K, by S( K) 410K. The degrees Kelvin is a function of the temperature in Celsius given by K(C)= C Find the speed of sound when the temperature is 30 degrees Celsius. Round to the nearest tenth.

8 HW#4 Name Unit 2 Functions HW #4 Algebra II Mrs. Dailey 1) A function is given by the set of ordered pairs {(2,5),(4,9),(6,13),(8,17)}. Write its domain and range in roster form. Domain: Range: 2) The function h(x) = x maps the domain given by the set {-2,-1,0,1,2} Which of the following sets represents the range of h(x)? (1) {0, 6, 10, 12} (3) {5, 6, 9} (3) {-5, 6, 9} (4) {1, 4, 5, 6, 9} 3) Which of the following values of x would not be in the domain of the function defined x 2 by f( x)? * Functions cannot be undefined x 3 (denominator = 0) (1) x = -3 (3) x = 3 (2) x = 2 (4) x = -2 4) Determine any values of x that do not lie in the domain of the function Justify your response. 3x 2 f( x). 2x 10 5) Which of the following values of x does lie in the domain of the function defined by g( x) 2x 7? (1) x = 0 (2) x = 2 (3) x = 3 (4) x = 5

9 6) Which of the following would represent the domain of the function y 6 2x? (Plug into y = in calculator. Look at graph and table of values) (1) {x: x >3} (3) {x: x 3} (2) {x: x <3} (4) {x: x 3} 7) The function y=f(x) is completely defined by the graph below. (a) Evaluate f(-4), f(3), and f(6). (b) State the domain and range of this function using interval notation. Domain: Range: 8) Which of the following represents the range of the quadratic function shown in the graph below? (1) (4, ) (3) (-,4) (2) (-,4] (4) [4, )

10 9) A child starts a piggy bank with $2. Each day, the child receives 25 cents at the end of the day and puts it in the bank. If A represents the amount of money and d stands for the number of days then A(d) = d gives the amount of money in the bank as a function of days (think about this formula). (a) Evaluate A(7), and A(30) (b) For what value of d will A(d) = $10.50 (c) Explain why the domain does not contain the value d= 2.5. (d) Explain why the range does not include the value A = $3.10.

11 HW#5 Name Algebra II Unit 2 Functions HW #5 Mrs. Dailey 1) Which of the following graphs illustrates a one-to-one relationship? 2) Which of the following graphs does not represent that of a one-to-one function? 3) In which of the following graphs is each input not paired with a unique output? (Fails horizontal line test) 4) In which of the following formulas is the variable y a one-to-one function of the variable x? (Hint Plug into y= and look at table and graph) (1) y = x 2 (2) y x (3) y = 2x (4) y = 5

12 5) Which of the following tables illustrates a relationship in which y is a one-to-one function of x? 6) Physics students drop a basketball from 5 feet above the ground and its height is measured each tenth of a second until it stops bouncing. The height of the basketball, h, is clearly a function of the time, t, since it was dropped. (a) Sketch the general graph of what you believe this function would look like. (b) Is the height of the ball a one-to-one function of time? Explain your answer.

13 HW#6 Name Algebra II Unit 2 Functions HW #6 Mrs. Dailey 1) If the point (-7,5) lies on the graph of y = f(x), which of the following points must lie on the graph of its inverse? (1) (5,-7) (2) 1 1, 7 5 (3) (7,-5) (4) 1 1, 7 5 Inverse: Switch x and y coordinates. 2) The function y=f(x) has an inverse function y = f -1 (x). If f(a) = -b then which of the following must be true? (1) f -1 (-b) = -a (3) f -1 (-b) = a (2) f a b (4) f -1 (b) = -a 3) The graph of the function y= g(x) is shown below. The value of g -1 (2) is (1) 2.5 (3) 0.4 (2) -4 (4) -1

14 Inverses only exist when functions pass horizontal line test (One-to-One). 4) Which of the following functions would have an inverse that is also a function? Inverse: Switch x and y coordinates. 5) For a one-to-one function it is known that f(0)= 6 and f(8) = 0. Which of the following must be true about the graph of this function s inverse? (1) its y-intercept = 6 (3) its x-intercept = -6 (2) its y-intercept = 8 (4) its x-intercept = -8 6) The function y = h(x) is entirely defined by the graph shown below. (a) Sketch a graph of y= h -1 (x). (b) Write the domain and range of y=h(x) and y = h -1 (x) using interval notation. y = h(x) y= h -1 (x) Domain: Domain: Range: Range:

15 7) The domain and range of a one-to-one function, y= f(x), are given below in set-builder notation. Give the domain and range of this function s inverse also in set-builder notation. y= f(x) y= f -1 (x) Domain: {x -3 x < 5} Domain: Range: {y y > -2} Range:

16 Name Algebra II Unit 2 Functions HW #7 Mrs. Dailey 1) The piecewise linear function f(x) is shown to the right. Answer the following questions based on its graph. (a) Evaluate each of the following based on the graph: (i) f(4) (ii) f(-3) HW#7 (b) How many solutions would exist to the equation f(x) = 2? Explain how you determined your answer. c) State the zeroes of f(x). (c) Over which of the following intervals is f(x) always increasing? (1) -7 < x < -3 (3) -5 < x < 5 (2) -3 < x < 5 (4) -5 < x < 3 (d) State the coordinates of the relative maximum and the relative minimum of this function. Relative Maximum: Relative Minimum: (e) Over which of the following intervals is f(x) < 0? (1) -7 < x < -3 (2) 2 x 7 (3) -5 < x < 2 (4) -5 x 2

17 2) Draw a graph of y = f(x) that matches the following characteristics. f(-8) = -5 and zeroes at x = -6, -2, & 3 Increasing on: -8 < x < -4 and -1 < x < 5 Decreasing on: -4 < x <-1 Absolute maximum of 7 and Absolute minimum of -5 3) A continuous function has a domain of -7 x 10 and has selected values shown in the table below. The function has exactly two zeroes and a relative maximum at (-4,12) and a relative minimum at (5, - 6). x f x (a) State the interval on which f(x) is decreasing. (b) State the interval over which f(x) < 0.

18 Unit 2 Test Review Name CC Algebra II Unit 2 Functions Date Mrs. Dailey Topics One-to-One (Vertical line test and Horizontal line test) Inverse Functions Relative Min/Max & Absolute Max/Min Graph is increasing Functions (vertical line test: every x-value one 1 y-value) Compositions f(g(x) Domain and range 1) Which of the following sets of ordered pairs would not be considered a function? (1) {(-4,1), (-1,7),(3,8),(5,3)} (2) {(-2,5),(6,1), (-2,10), (6,-1)} (3) {(2,8), (4,10), (6,8), (8,10)} (4) {-3,5),(3,-5), (-6,7), (6,-7)} 2) In the following graph, the height of an object, in feet, is given as a function of time in seconds. Which of the following would be the range of this function? (1) [0,5] (2) [0,11] (3) [176,576] (4) [0,576] 3) In which of the following four graphs is the output not a function of the input? (1) (2) (3) (4)

19 4) If 1 f ( x) x 6, then which of the following values solves the equation f(x) = 10? 2 (1) 1 (3) -8 (2) -4 (4) 11 5) The function f is defined by the formula f(x)= x and the function g is defined by the graph shown below. Which of the following is the value of f(g(2))? (1) 18 (3) 5 (2) 14 (4) 9 g x 6) Given the function f(x) shown in the graph below, for which of the following intervals is f(x) > 0? (1) (0,8) (2) [0,8] (3) (-4,6) (4) [-4,6] 7) Which of the following values of x would not be in the domain of the function x 7 f( x)? 2x 5 (1) 7 (3) 5 1 (2) 2 2 (4) 7

20 8) If the point (4,-2) lies on the graph of y= f(x), then which of the following points must lie on the graph of its inverse, i.e. y = f -1 (x)? (1) (-2,4) (3) (-4,2) 1 1 (2), 4 2 (4) (4,2) 9) Which of the following graphs represents a one-to-one function? (1) (3) (2) (4) 10) Given the function shown below, over which of the following intervals is the function always increasing? (1) 0 < x < 5 (2) -5 < x < 2 (3) -1 < x < 4 (4) -9 < x < -5

21 11) Which of the following is the y-intercept of the piecewise defined function 6x 5 x 2 gx ( )? 2 ( x 3) 1 x 2 (1) 5 (3) -1 (2) 6 (4) 8 12) Given the function y=f(x) shown graphed below, answer the following questions. (a) State the value of f(2). (b) How many values solve the equation f(x) = 5? Explain how you arrived at your answer. (c) On the interval 0 < x < 4 is the function increasing or decreasing? How can you tell? (d) If a second function is defined by the formula value of g(f(1))? Justify. 2x 7 gx ( ), then what is the 3 (e) Is the inverse of this function a function? Explain.

22 13) For the function f ( x) x 9, either x = 0 or x = 45 is a member of its domain. Determine which and explain how you arrived at your answer. 14) Is the function y x 6 2 a one-to-one function? Explain. 15) Given the function y = f(x) shown below do the following. (a) Graph the function s inverse, f -1 (x). (b) State the range of f -1 (x). (c) What is the value of f -1 (-3)?

23 16) Two functions are given below in table and equation form. Using these functions, give the value of f(g(-4)). Show how you arrived at your answer. x f(x) x g x 9