Note: Always give exact answers and always put your answers in interval notation when applicable.

Transcription

1 Note: Always give exact answers and always put your answers in interval notation when applicable.

2 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. (ie, y is NOT a function of y)

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 Example 1: Does the following graph represent a function?

5 Example 2: Does the following graph represent a function?

6 Example 3: Do the following relations represent functions? (a) Fido Bossy Silver Frisky Polly (b) Civil War WWI WWII Korean Vietnam (c) x y

7 The set of all possible x-values is defined as the domain. The set of all resulting y-values is defined as the range. *Always write your answers in interval notation unless instructed otherwise

8 Example 4: Solve the equation for y to determine if it represents y as a function of x. If so, determine the domain and range. 3y 3x 2 = 12x + 9

9 (a) Example 5: Find the domain and range of each of the following functions. (b) (c)

10 Graphs of Relations and Functions

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

12 Make a table listing ordered pairs that satisfy the following equation. Then Graph the equation using the ordered pairs. y = 1 x 2 X Y

13 Graph the following equation. Is it 1-1? What is the Domain and Range? y = x 3

14 Determine if the following relations describe y as a function of x. Graph each relation. (a) y = 16 x 2 (b) y = 16 x 2 (c) y 2 = 16 x 2

15 (a) Graph each relation. Determine the intervals when the following relations are increasing, decreasing, and constant. (b) (c)

16 Transformations

17 There are 2 categories of transformations. 1. Rigid Transformations 2. Nonrigid Transformations

18 There are 3 different rigid transformations: 1. Vertical Shifts up and down 2. Horizontal Shifts left and right 3. Reflection Reflects over an axis

19 f(x) + a is f(x) shifted upward a units f(x) a is f(x) shifted downward a units f(x + a) is f(x) shifted left a units f(x a) is f(x) shifted right a units f(x) is f(x) flipped upside down ("reflected about the x-axis")

20 (a) (b) How many units is each function shifted? In which direction? (c)

21 (a) (b) (c) How many units are each graph shifted? In which direction.

22 There are 2 types of nonrigid transformations. 1. Stretching - Let a > 1. Then y = af(x) stretches the graph by a factor of a. 2. Shrinking - Let 0 < a < 1. Then y = af(x) shrinks the graph by a factor of a.

23 Graph the following equations on your calculator.

24 Use transformations to graph the following function. State the domain and range. 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).

25 (a) Describe the transformation in words. Then verify by graphing on your calculator. (b)

26 Operations With Functions

27 Provided that g(x) 0.

28 Let following:. Evaluate the (a) f + g (b) f g (c) f g (d) f/g

29 Let following:. Evaluate the (a) y - w (b) y/w

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

31 Let following:. Evaluate the (a) (f g)(x) (b) (g f)(x)

32 Inverse Functions

33 ***A function has an inverse if and only if the function is 1-1.***

34 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)

35 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 f -1 and vice versa.

36 Find the equation of the inverse of f(x) =2x-3

37 Graph the inverse of the following function: f x = x *Remember: reflect the graph of f(x) over the line y=x to get the graph of the inverse.

38 Find the inverse of the following function. x y

39 Determine if it s a function Graphs of functions Finding Domain and Range Operations of Functions Transformations Functions and their Inverses