MAT116 Final Review Session Chapter 2: Functions and Graphs

Transcription

1 MAT116 Final Review Session Chapter 2: Functions and Graphs Note: Always give exact answers and always put your answers in interval notation when applicable.

2 Section 1 If the value of x determines the value of y, we say that y is a function of x. If there is more than one value of y corresponding to a particular x-value then y is not determined by x. (i.e., y is NOT a function of x)

3 Vertical Line Test A graph is a graph of a function if and only if there is no vertical line that passes through the graph more than once.

4 Examples: Do these represent a function? 1. 2.

5 Examples: Do these relations represent a function? Fido Bossy Silver Frisky Polly Civil War WWI WWII Korean Vietnam x y

6 Domain and Range The set of all possible x-values is defined as the domain. The set of all resulting y-values is defined as the range.

7 Examples: Determine if the following are functions, state their domain and range. 6. 3y 3x 2 = 12x y = 3x y = x 3 9. y = 3 x

8 1-1 A function is 1-1 if and only if it s graph passes the vertical line test AND the horizontal line test.

9 Examples: Graph the function, state if it is 1-1, the domain and range 10. y = 1 x y = x 3

10 Examples: Determine if the following equations are functions. 12. y = 16 x y = 16 + x y 2 = 16 x 2

11 Circles A graph of any equation of the form (x h) 2 + (y k) 2 = r 2 is a circle with center (h, k) and radius r. Circles do not represent a function. Examples: What is the center and radius of the following circle? 15. (y 2) 2 +(x + 4) 2 = 16

12 Examples: Determine where the graph is increasing, decreasing or constant. 16.

13 Transformations There are two categories of transformations: Rigid Transformations Nonrigid Transformations

14 Rigid Transformations There are 3 different rigid transformations: 1. Vertical Shifts up and down f(x) + a is f(x) shifted upward a units f(x) a is f(x) shifted downward a units 2. Horizontal Shifts left and right f(x + a) is f(x) shifted left a units f(x a) is f(x) shifted right a units 3. Reflection reflects over and axis f(x) is f(x) flipped upside down (reflected over x-axis)

15 Examples: How many units is each function shifted? In which direction? 17. h x = x f x = x g x = x n x = (x + 2) q x = x p x = (x 7) 5 +3

16 Nonrigid Transformations There are 2 types of nonrigid transformations. 1. Stretching Let a > 1. Then y = a f(x) stretches the graph by a factor of a. 2. Shrinking Let 0 < a < 1. Then y = a f(x) shrinks the graph by a factor of a. * All the y-coordinates on f(x) are multiplied by a, so the graph stretches or shrinks in the y direction.

17 Examples: Graph the following on your calculator. 23. y = x y = 1 3 x2 25. y = 5x 2

18 Examples: Use transformations to graph the following function. State the domain and range. 26. y = x Note: Be sure to follow the order of operations while translating the function. Please Excuse My Dear Aunt Sally. (Parentheses, exponents, multiplication/division, addition/subtraction).

19 Examples: Describe the transformation in words. 27. t x = 2 x f x = x 3 1 2

20 Operations with Functions f + g x = f x + g x f g x = f x g x f g x = f x g x f/g x = f(x)/g(x) where g(x) 0

21 Examples: Evaluate the following. Let y x = 2x 2 3 and w x = 2x (y + w)(1) 30. (w y)(2) 31. (y w)(4) 32. y/w(x)

22 Composition If f and g are two functions, the composition of f and g, written f g, is defined as follows:

23 Examples: Evaluate the following. Let f x = x 2 1 and g x = 3x (f g)(x) 34. (g f)(x)

24 Inverse Functions A function has an inverse if and only if the function is 1-1. The inverse of a one-to-one function f(x) is the function f 1 such that: Note: The domain of f(x) is the range of f 1 (x) The range of f(x) is the domain of f 1 (x)

25 To find the inverse of a function f(x): 1) Replace f(x) with y 2) Interchange x and y 3) Solve the equation for y. 4) Replace y with f 1 (x). 5) Verify that D f = R 1 f and vice versa.

26 Examples: Find the equation of the inverse. 35. f x = 2x 3

27 Examples: Graph the inverse of the following function: 36. f x = x 2 + 6x + 9; x 3 Remember: reflect the graph of f(x) over the line y = x to get the graph of the inverse.

28 Examples: Find the inverse. 37. x y

29 Chapter 2 Review Determine if it s a function Graphs of functions Finding Domain and Range Operations of Functions Transformations Functions and their Inverses

30 Example Answers 1) A function 2) Not a function 3) Not a function 4) A function 5) A function 6) Domain: (, ), Range: [ 1, ) 7) Domain: (, ), Range: (, ) 8) Domain: (, ), Range: [ 3, ) 9) Domain: (, 3], Range: [0, ) 10) Domain: (, ), Range: (, 1] 11) Domain: [0, ), Range: [ 3, ) 12) Yes 13) Yes 14) Not a function 15) Center = (2, -4) Radius = 4 16) Increasing on [ 3, 1] [0,2] Decreasing on [2,3] Constant [ 1,0] 17) 2 units right 18) 2 units down 19) 5 units up 2 20) 2 units left 21) 5 units right 2 22) 7 units right, 3 units up 23 25) Graph 26) D: (, ) R: (, 1] 27) Magnified by 2, moved right 2 units, up 4 units 28) Flipped over x-axis, moved 3 units right, move down ½ units 29) 5 30) 3 31) ) x 2 33) 9x 2 24x 17 34) 3x ) y = x ) Graph