PreCalculus FUNctions Unit 1 Packet

Transcription

1 Name Hr VOCABULARY Function: Intercepts: Increasing: Decreasing: Constant: Continuous: Even: Odd: Local Maximum: Local Minimum: Discussion: Possible or Not? EXAMPLE 1: Increasing interval(s): Decreasing interval(s): Intercepts: Local Max/Min: For which interval(s) is f(x) > 0? f(x) < 0? EXAMPLE 2: Increasing interval(s): Decreasing interval(s): Constant interval(s): Intercepts: Local Max/Min: For which interval(s) is f(x) > 0? f(x) < 0? 1

2 EXAMPLE 3: f( x) 2 x 2 x 4 Is the point 3 1, 5 on the graph of f? If x = 2, what is the value of f(x)? Evaluate f(3 a). Find the domain of f. List the x and y-intercepts. Plug the function into your calculator. List the interval(s) where f is increasing. Is f a continuous function? Justify your answer. For which interval(s) is f(x) < 0? State the x-coordinates of the local max and local min, if any. State the range of the function. Piecewise Functions: A function is defined uniquely on different parts of its domain. EXAMPLE: A cell phone plan costs $80 a month. The plan includes 400 free minutes and charges 20 for each additional minute of usage. We could represent the monthly charges as a function of the number of minutes used with a piecewise function. 2

3 Graph the following piecewise functions: f(x) = { 3 if x < 2 x 1 if x 2 g(x) = { x2 if x 1 2x + 1 if x > 1 Common Functions: Equation Graph Properties Transformation Shift Up 3 units. Evaluate f( 3). Even, Odd, or Neither? Generalize: Evaluate f(2 + b). Shift to the left 2 units. Even, Odd, or Neither? Generalize: 3

4 Evaluate f ( 2 3 ). Even, Odd, or Neither? Reflect the graph across the y-axis. Generalize: Evaluate 16 f. Even, Odd, or Neither? Reflect the graph across the x-axis. Generalize: Evaluate f 5 7. Shift the graph to the right 5 units and down 2. Even, Odd, or Neither? Evaluate f( 3). Stretch the graph by a factor of 2. Even, Odd, or Neither? Generalize: Evaluate f ( π 3 ). Compress the graph by a factor of ½, shift to the left 1 and up 2. Even, Odd, or Neither? Generalize: 4

5 Building Functions from Functions (Function Composition): f ( g( x)) f g Knowing how a function is put together is an important step when applying the tools of calculus. Use the words input and output, as appropriate, to fill in the missing blanks: The function f(g(x)) uses the of the function g as the to the function f. The function g(f(x)) uses the of the function f as the to the function g. EXAMPLE: Let f ( x) 2 x and g( x) x 1. f(1) + g(3) = f(x) + g(x) = f( 2) g(8) = f(g(3)) = g(f(x)) = g(f(2)) = f(f( 5)) = g(g( 1)) = f(g(x)) = EXAMPLE: Given that h(x) = f(g(x)), fill out the table of values for h(x). x f(x) g(x) h(x) EXAMPLE: g(f(2)) = f(g(0)) = g(g( 2)) = f(f(1)) = f(x) g(x) 5

6 Exploring inverses graphically: Graph f ( x) Exploring One-to-One and Inverse Functions: x and Graph the line y x. 2 g( x) x, x 0. How are the graphs of f and g related to the line y x? EXAMPLE 1: Graph f ( x) 2x 3 and gx ( ) x 3 2 x f( x ) x gx ( ) Find and simplify f ( g( x )). Find and simplify g( f ( x )). EXAMPLE 2: Graph f x 3 ( ) x 2 and g x 3 ( ) x 2 x f( x ) x gx ( ) Find and simplify f ( g( x )). Find and simplify g( f ( x )). Conclusion: If two functions are inverses, then f ( g( x)) g( f ( x)). 6

7 EXPLORATION: Find the inverse of the following function: Super Bowl XLI XLII XLIII XLIV XLV XLVI XLVII MVP Peyton Manning Eli Manning Santonio Holmes Drew Brees Aaron Rodgers Eli Manning Joe Flacco Is the inverse relation a function? Why or why not? Find the inverse relation for the following function: {( 1,3),(1,7),(2,5),(4, 3)} Is the inverse relation a function? Why or why not? What quality must a function have if its inverse is going to be a function? Recall that a graph must pass the test to be a function. So, a one-to-one function must pass BOTH the test and the test. Determine whether the following functions are one-to-one: f x 3 ( ) x 2 f ( x) x f ( x) ( x 2) 2 7

8 Three steps to find an inverse algebraically: EXAMPLES: Find the inverse of the following functions. f ( x) 3x 2 5 x 3 f( x) 2 f ( x) x 2 Modeling with Functions: Find a MODEL for these situations: 1. A poster is 10 inches longer than it is wide. Find a function that models its area in terms of its width. 2. The height of a cylinder is four times its radius. Find a function that models the volume of the cylinder in terms of its radius. 3. Find a function that models the radius of a circle in terms of its area. 4. The junior class has paid $500 to a disc jockey for a dance. Tickets for the dance are $20 each. Express the net income as a function of the number of tickets sold. 8

9 EXAMPLE 1: Modeling the Volume of a Box General Mills packages its cereal boxes with the following proportions: Its width is 3 times its depth and its height is 5 times its depth. EXAMPLE 2: Fencing a Dog Pen Suppose you have 140 feet of fencing to make a pen for your dog. Find a function that models the area of the dog pen you can fence. a) Find a function that models the volume of the box in terms of its depth. b) Find the volume of the box if the depth is 1.5 in. EXAMPLE 3: Suppose you wish to get Mrs. Sapp a nice gift. Instead of buying a gift box, you want to impress her by constructing a rectangular box from a piece of cardboard 16 inches wide and 21 inches long by cutting congruent squares from each corner and then bending the sides. Model the volume of the box as a function of the size of a corner square. 9

10 PRACTICE: 1. Graph: f a) f (0) Questions: What is this value called? b) Find f ( 2). c) List the interval(s) where f is increasing. d) For what interval(s) is f(x) > 0? e) For how many x-value(s) does f(x) = 3? f) List the solution(s) to the function f. What are these value(s) called? g) Is the function continuous? Explain. 2. a) Approximate the y-intercept. f b) Find f (2). c) List the interval(s) where f is decreasing. d) For what interval(s) is f(x) < 0? e) State the domain and range. f) State the x-value of the local maximum. g) For what number(s) is (x) = 0? h) Is the function continuous? i) Is the function one-to-one? 10

11 3. Graph y = 3sin x for 2π x 2π. a) Evaluate f ( π 2 ). b) List the interval(s) where f is increasing. c) Is the function continuous? If not, state the x- value(s) where the function is discontinuous. d) State the domain and range of h. e) If f even, odd, or neither? Explain. f) Is the function one-to-one? Use the given equation to answer the questions. 4. x 2 gx ( ) x 6 a) Is the point 3,14 on the graph of g? Explain. b) If x 4, what is gx ( )? c) If gx ( ) 2, what is x? d) State the domain: e) List the y-intercept(s), if any: f) List the x-intercept(s), if any: 11

12 5. f( x) 4x x 3 a) Is the point 3,0 on the graph of g? Explain. b) Is the function continuous? Why or why not? c) Evaluate f (7). d) State the domain: e) List the y-intercept(s), if any: f) List the x-intercept(s), if any: 6. Graph the piecewise function: 3 x if x 1 hx ( ) 2x 3 if x 1 a) State the x-intercept(s). b) Find h( 2). c) List the interval(s) where h is constant. d) List the interval(s) where h is decreasing. e) State the x-value of the local minimum on the interval 0,2. f) Is the function continuous? If not, state the x-value(s) where the function is discontinuous. g) State the domain and range of h. h) Is the function one-to-one? 12

13 7. Graph the piecewise function: sin 2 x 2 x 0 f( x) 2cos x 0 x2 a) State the x-intercept(s). b) Find f ( ). c) List the interval(s) where f is increasing. d) State the x-value of the local minimum on the interval 0,2. e) Is the function continuous? If not, state the x-value(s) where the function is discontinuous. f) State the domain and range of f. g) Is the function one-to-one? 8. Watch the YouTube video: Write down the piecewise model for Cost from the Friendly Energy Company example. 9. Westar Energy charges its electric customers a base rate of $6.00 per month, plus $0.10 per kwh (kilowatt hour) for the first 300 kwh used and $0.06 per kwh for all usage over 300 kwh. Suppose a customer uses x kwh of electricity in one month. a) Express the monthly cost C as a function of x. b) Graph the function between 0 x

14 Describe how to transform the graph of f into the graph of g. 10. f(x) = x + 2 g(x) = 5 + x f(x) = (x 1) 3 g(x) = (x 1) Each function below is a transformation of f ( x) x. Write a formula for each function. 12. g(x) = 13. h(x) = 14. k(x) = 15. r(x) = 14

15 16. Refer to the graph of f below. Sketch each new function. f a) y = f(x 1) b) f(x) + 1 c) f(2x) d) f( x) 15

16 17. Let a) f x 2 ( ) x 4 f( x) gx ( ), g( x) x 1, and hx ( ) x x. Evaluate: 1 b) h( x) g( x) State the domain: State the domain: c) f ( g( x )) d) g( f (3)) e) gh ( (0)) f) ( f f)( 2) 18. Use the functions below to evaluate. f g a) f( g( 2)) b) ( g f)(1) c) f( f(2)) d) ( f g)(0) e) Is f a one-to-one function? f) Is g a one-to-one function? 16

17 19. Find the inverse of f ( x) 2x 4 and graph f and its inverse. 20. Find the inverse of f and its inverse. f x 3 ( ) x 4 and graph State the domain of: f : 1 f : State the domain of: f : 1 f : 21. The formula F(C) = 1.8C + 32 converts Celsius temperature to Fahrenheit. Find a formula for the inverse function, giving Celsius as a function of Fahrenheit. Explain the meaning of F 1 (200) = The formula V = 4 3 πr3 gives the volume of a sphere of radius r. Find a formula for the inverse function giving radius as a function of volume. Explain the meaning of f 1 (10) = Suppose we have 2400 feet of fencing and want to fence off a rectangular field that borders a straight river. (We don t need fencing along the river.) a) Make a sketch of the situation: b) Find a function that models tha area of the field in terms of one of its sides. 17

18 24. A rectangle is inscribed under one bump of the cosine curve, with two vertices on the x- axis and the other two vertices on the curve y cos x where x 2 2 a) Make a sketch of the situation: b) What is the area of the rectangle, given in terms of the x-coordinate of one of the vertices on the x-axis? 25. Take the Functions Quiz online and record your answers here: EXPANSION 1: Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum. EXPANSION 2: Some functions have the property that f(a + b) = f(a) + f(b) for all real numbers a and b. Which of the following functions have this property? Show your work. a) h(x) = 2x b) g(x) = x 2 c) f(x) = 5x 2 d) H(x) = 1 x 18