Warm-up (10 min. – No Talking) Sketch the graph of each of the following function. State the domain and range. Describe how and to which basic function it relates. 1) y = sin x - 4 2) y =  x-2 3) y = [x – 1] = int (x-1) 4) y = x ) y = -|x| + 2

Warm-up (Answers) Do Graphs on Calculator DomainRangeTo Basic? 1) y = sin x - 4 (- ,  ) [-5,-3]Sine down 4 2) y =  x+2[2,  )[0,  ) Square root left 2 3) y = [x – 1] = int (x-1) (- ,  ) I Greatest Integer right 1 4) y = x (- ,  )[3,  ) Squaring up 3 5) y = -|x| + 2 (- ,  )(- ,2] Abs. Value reflected over x-axis & up 2

Homework Check You have 5 min. to write your answers to the following: #3,15, 21, 30, 39 Exchange papers for gradinggrading Give a score out of 10 and initial. Return to owner for reviewing. Pass to your left (my right) for recording. Questions on the homework? Volunteers?

Homework Answers 3) j6) f9) d12) b15) Ex. 7,8 18) Ex. 3,4,11,1221) y = x 2, y = 1/x, y = |x| 24) y = x, y = x 3, y = ln x 27) y = x, y = x 3, y = 1/x, y = sin x 30) D: (- ,  )R: [0,  ) 33) D: (- ,  )R: All Integers 36) a) Inc: [(2k-1)  /2, (2k+1)  /2] Dec: [(2k+1)  /2, (2k+3)  /2] b)Neither c) Min. of 4 at (2k-1)  /2 Max. (2k+1)  /2 d) Sine shifted 5 units up

Homework Answers 39)a) Inc. [0,  ); Dec. (- , 0]b) even c) Min. of -10 at x = 0 d) Abs. Value shifted 10 units down 42) a) Inc. (- ,0]; Dec. [0,  )b) even c) Max. of 5 at x = 0 d) Abs. Value reflected across x-axis & then shifted 5 units up 45 – 54See transparency of graphs

Building Functions from Functions To build functions from old functions Use basic operations (addition, subtraction, multiplying and dividing) to build new functions Determine new functions formed from composition or inverse operations

Why are building functions important? If a person has one investment account whose interest is given by f(t) = 250t and another account whose interest is given by g(t) = 420t, how can you write a function h(t) to give the total interest?

Activity #1 Use f(x) = x 2 and g(x) = x to compare the graphs of the following functions. Discuss the domain, range, and zeros of each pair of functions. 1) (f+g)(x)f(x) + g(x) 2) (f-g)(x)f(x) – g(x) 3) (fg)(x)f(x)g(x) 4) (f/g)(x)f(x)/g(x) g(x)  0 What do you notice?

Definitions of Basic Operations on Functions Let f and g be two functions with intersecting domains. Then for all values of x in the intersection, the algebraic combinations of f and g are defined by the following rules Sum(f+g)(x) = f(x) + g(x) Difference(f-g)(x) = f(x) – g(x) Product(fg)(x) = f(x)g(x) Quotient (f/g)(x) = f(x)/g(x), provided g(x)  0 In each case, the domain of the new functions consists of all numbers that belong in BOTH the domain of f and of g. As noted, the zeros of the denominator are excluded from the domain of the quotient.

Ex1 Let f(x) =  x-1 and g(x) = x 2. Find formulas for f + g, f- g, fg, f/g, and ff. Give the domain of each. New FunctionsDomain f + g = f – g = fg = f/g = ff =

Composition of Functions Let f and g be two functions such that the domain of f intersects the range of g. The composition f of g, denoted f  g, is defined by the rule (f  g)(x) = f(g(x)) The domain of f og consists of all x-values in the domain of g that map to g(x)-values in the domain of f.

Checking the domain of a composite x g(x) f(g(x)) g f f  g

Ex 2 Let f(x) = 3x + 2 and g(x) = x – 1. Find f  g and g  f. State the domain of each.

Tonight’s Assignment P Ex #3-24 m. of 3 Review 12 basic functions

Ex3 Let f(x) = x 2 -1 and g(x) =. Find the domain of f(g(x)) and g(f(x)).

Decomposing Functions Reversing a composition of two functions is not always as easily visible as finding the composition of two function.

Ex4 For each function h, find the functions f and g such that h(x) = f(g(x)). (a) h(x) = (b) h(x) = (x + 1) 2 – 3(x + 1) + 4

One to One A function is one-to-one if and only if it passes both the VLT and the HLT. Another way: if both compositions are equal to x once you simplify.