ch1.1 – functions functions – for every element x in a set a function machine there is exactly...

Ch1.1 – FunctionsFunctions – for every element x in a set A function machine

there is exactly one element y from set B xthat corresponds to it.Set A – (the set of inputs) is the domain ySet B – (set of outputs) is the range

Ex1) A = {a,b,c}, B = {1,2,3,4,5}. Which of following are functions from A to B?a) {(a,2),(b,2),(c,4)} b) {(a,4),(b,5)}

c) d)

f(x)

abc

12345

abc

12345

y = x2

dependent independent variable variable

f(x) = x2

Ex2) Which represent y as a function of x?

a) x2 + y = 1 b) –x + y2 = 1

Ex3) Let g(x) = -x2 + 4x + 1 Solve:

a) g(2) b) g(t) c) g(x+2)

Ex4) Evaluate the piecewise function for x = -1,0,1

x2 + 1 x < 0x – 1 x > 0

f(x) =

Ex5) Find the domain of each:

a) f:{(-3,0),(-1,4),(0,2),(2,2),(4,1)}

ChP.1A p92 1-7odd,25-41odd (just a and c)

xxhd

rV

xxgb

4)( )

3

4 c)

5

1)( )

3

Ch1.1A p92+ 1-7odd,25-41odd

ChP.1B – More Functions

Ex1) Find all real values for x such that f(x) = 0 in:

Ex2) Find the values where f(x) = g(x):

f(x) = x2 + 2x + 1 g(x) = x + 2

5

43)(f

x

x

Ex3) For f(x) = x2 – 4x + 7, find

h

xfhx )()(f

HW#75) For g(x) = 3x – 1 find:

HW#76)

Ch1.1B p92+ 26-40even, 43-59odd,71-75odd

3 ,3

)3()(g

xx

gx

1 t,1

)1()(f find

1)(

t

gt

ttf

Ch1.1B p92+ 26-40even,43-59odd,71-75odd

Ch1.2 – Graphing Functions 4Ex1) The graph of function f is shown. 3 a) Find the domain. 2 b) Find the values of f(-1) and f(2) 1 c) Find the range of f.

-4 -3 -2 -1 -1 1 2 3 4 5-2-3-4-5

(-1,-5)

(2,4)

(4,0)

Ex2) Use the vertical line test to determine which graphs represent y as a function of x.

a) b) c)

Increasing Function – if x1 < x2 , then f(x1) < f(x2)Decreasing function – if x1 < x2 , then f(x1) > f(x2)Constant function – for all x, f(x1) = f(x2)

Ex3) For the following graphs determine where the functions are increasing,decreasing, and constant. t + 1 t < 0a) f(x) = x3 b) f(x) = x3 – 3x c) f(t) = 1 0 < t < 2

-t + 3 t > 2

SymmetryA function f is even if for each x: f(x) = f(-x)A function f is odd if for each x: f(-x) = -f(x)

symmetry to y-axis symmetry to origin symmetry to x-axis

Ex4) Determine whether each function is odd, even, or neither:a) g(x) = x3 – x b) h(x) = x2 + 1 c) f(x) = x3 – 1

Ch1.2 p105+ 1-13odd,19,21, 25-33odd,37-45odd

Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd

Ch1.3 – Graphs of FunctionsThe most common graphs in algebra:

f(x) = c f(x) = x f(x) = |x| Constant function Identity function Absolute value function

f(x) = f(x) = x2 f(x) = x3

Square root function Square function cube function(works w all (works w all even powers) odd powers)

x

Shifts: Shift upward: f(x) + c Shift downward: f(x) – c Shift right: f(x – c) Shift left: f(x + c)

Ex1) How does each function compare to f(x) =x3

a) g(x) = x3 + 1 b) h(x) = (x – 1)3 c) k(x) = (x + 2)3 + 1

Ex2) Find eqns for each function: f(x) = ___ g(x) = ___ h(x) = ___

g(x)

h(x) f(x)

654321

-1-2-3-4-5-6

-6 -5 -4 -3 -2-1 1 2 3 4 5 6

Reflections: f(x) = x2

Reflection to the x-axis:

Reflection to the y-axis

Ex3) Find eqns for each function that is a transformation of f(x) =x4

a) g(x) = b) h(x) =

Ch1.3A p116+ 1-9odd 13,19,23

Ch1.3A p116 1-9odd,13,19,23

Ch1.3B – More GraphsEx4) Graph each:

a) f(x) = – b) g(x) = c) h(x) = –

x x 2x

Nonrigid Transformations – stretch and shrink graphs

Ex5) Compare each function to f(x) = |x|

g(x) = 3|x| h(x) = x3

1

Ex6) Use a calculator to graph:

g(x) = 5(x2 – 2) h(x) = 5x2 – 2

Ch1.3B p116+ 15,17,21,25-35odd,41,43

Ch1.3B p116+ 15,17,21,25-35odd,41,43ID the common function and what transformation is shown

Ch1.3B p116+ 15,17,21,25-35odd,41,43

Ch1.1 – 1.3 Mid Chapter Review Ch1.1 p92+ 52,54,58,72,74 Ch1.2 p105 2,4,6,10,20,22,28,30 Ch1.3 p116 2,4,10,16,18,20,

26-36even

Ch1.1 p92+ 52,54,58,72,74

Ch1.2 p105 2,4,6,10,20,22,28,30

Ch1.3 p116 2,4,10,16,18,20,26-36even

Ch1.4 – Combinations of Functions

1. Sum: (f + g)(x) = f(x) + g(x)

2. Difference: (f – g)(x) = f(x) – g(x)

3. Product: (f.g)(x) = f(x).g(x)

4. Quotient: 0)( )(

)()(

xg

xg

xfx

g

f

Ex1) Find (f + g)(x) for the functions f(x) = 2x+1 and g(x) = x2 + 2x – 1 for x = 2.

Ex2) Find (f – g)(x) for the functions f(x) = 2x+1 and g(x) = x2 + 2x – 1 for x = 2

Ex3) Find and

for the functionsand list their domains.

)(xg

f

)(x

f

g

24)( ,)( xxgxxf

HW#s: 11, 21, 23 in class

Ch1.4A p126 5 -19odd

HW#s: 11, 21, 23 in class

Ch1.4A p 139 5 – 19 odd, (11,21,23 in class)

Ch1.4B – Composition of Functions

Ex4) Find for f(x) = x > 0

g(x) = x – 1 x > 1

Solve for and if possible.

))(()( xgfxgf

)(xgf x

)2(gf )0(gf

Ex5) Given f(x) = x + 2 and g(x) = 4 – x2

find and )(xgf )(xfg

Ex6) Given f(x) = x2 – 9 and g(x) =

find and )(xgf )(xff

29 x

Ex7) Given f(x) = 2x + 3 and g(x) = ½(x-3)

find and )(xgf )(xfg

Ex8) Express h(x) =

as a composition of two functions.

2)2(

1

x

Ex9) The number of bacteria in your food once pulled from the fridge is given by

N(T) = 20T2 – 80T +500 2 < T < 14, where T is temp in ˚C.When food is removed from fridge, it temp changes with time (t) by:

T(t) = 4t + 2 0 < t < 3, t in hours.

a) What does the composite N(T(t)) represent?b) What is the # of bacteria at t = 2hrs?c) At what time does the bacteria count reach 2000?

Ch1.4B p128 35-43odd, 49-53odd,61,63

Ch1.4B p128 35-43odd, 49-53odd,61,63

Ch1.5 – Inverse Functions

Domain Range

Range Domain

1234

5678

Ch1.5 – Inverse Functions

Domain Range f(x) = x + 4

Range Domain f -1(x) = x – 4 Inverse functions have the effect of undoing each other.

By defn, the domain of f must equal the range of f -1.Check with and

1234

5678

)(1 xff )(1 xff

Ex1) Find the inverse of f(x) = 4x

and verify that and equal the identity function.

)(1 xff )(1 xff

Ex2) Find the inverse of f(x) = x – 6

and verify that and equal the identity function.

)(1 xff )(1 xff

Ex3) Show that these functions are inverses:

f(x) = 2x3 – 1 3

2

1)(

xxg

Graphs of inverses are reflections around the y = x line.

Ex4) Graph Ex3 functions:

f(x) = 2x3 – 1

Ch1.5A p139+ 5-10 all, 11-17odd

3

2

1)(

xxg

Ch1.5A p139+ 5-10 all, 11-17odd

Ch1.5B – One-to-one FunctionsNot all functions have an inverse. To have an inverse a function must be

one-to-oneTo test, graph func on calc. If passes vertical and horizontal line tests,

then it has an inverse.

Ex5) Have inverses?

f(x) = 2x3 – 1 g(x) = x2 – x

Ex6) Find the inverse of

Procedure:1. Check w calc that it has inverse.

2. Replace f(x) with y.3. Interchange x and y4. Solve for y.

5. If it works, call it f -1(x).

2

35)(

xxf

Ex7) Find the inverse ofand sketch.

Ch1.5B p139 21,22,26,28,41-55odd

32)( xxf

Ch1.5B p139 21,22,26,28,41-55odd

Ch1 Rev p152 1 – 63 odd

654321

-1-2-3-4-5-6

-6 -5 -4 -3 -2-1 1 2 3 4 5 6

ch1.1 – functions functions – for every element x in a set a function machine there is exactly...

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