Functions ns – for every element x in a set A function machine e is exactly one element y from set B x corresponds to it. A – (the set of inputs) is the domain B – (set of outputs) is the range = {a,b,c}, B = {1,2,3,4,5}. h of following are functions from A to B? (a,2),(b,2),(c,4)} b) {(a,4),(b,5)} d) f(x) a b c 1 2 3 4 5 a b c 1 2 3 4 5
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Ch1.1 – Functions Functions – for every element x in a set A function machine there is exactly one element y from set B x that corresponds to it. Set A.
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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)
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4)( )
3
4 c)
5
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Ch1.1A p92+ 1-7odd,25-41odd
Ch1.1A p92+ 1-7odd,25-41odd
Ch1.1A p92+ 1-7odd,25-41odd
Ch1.1A p92+ 1-7odd,25-41odd
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
Ch1.1B p92+ 26-40even,43-59odd,71-75odd
Ch1.1B p92+ 26-40even,43-59odd,71-75odd
Ch1.1B p92+ 26-40even,43-59odd,71-75odd
Ch1.1B p92+ 26-40even,43-59odd,71-75odd
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
Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd
Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd
Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd
Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd
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
Ch1.3A p116 1-9odd,13,19,23
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