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Welcome to MM250
Unit 5 Seminar:
Functions and Graphs
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Left mouse click and hold.Drag to the right to enlarge the pod.
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Line y = 2x + 1
Function f(x) = 2x + 1
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x -------------> ---------> f(x) f
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Evaluating functions
Ex: g(x) = 3x + 2
Then g(0) = g(1) = g(b) = g(x+4) =
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Different types of Functions
f(x) = x2 + 1
f(x) = | x |
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Different types of Functions
Piecewise functions
Ex: f(x) = x2 when x <0 = 5x -1 when 0 ≤ x < 3
= 2x when x ≥ 3
What is f(-1), f(2), f(3)?
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Relation
a relation is a set of ordered pairs
Ex: { (1,0), (3,2), (1,5), (6,8), (7,2) }
Domain is the set of first elements {1, 3, 6, 7}Range is the set of second elements {0, 2, 5, 8}
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Functions are special relations
Ex: { (1,2), (1, 3), (3, 4), (7,8) } not a function
Ex: { (1,3), (2,3), (5,7), (6,7) } function?
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y = x2 is a function
every x goes to one y
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x = y2 does not define y as a function of x
one x goes to two y's
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f(x) = x4 - 5x2 + 4
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Odd and even functions
A function is even is f(-x) = f(x) ex: f(x) = x2
A function is odd if f(-x) = -f(x) ex: f(x) = x3
A function is neither of none of the above apply
g(x) = x5 - x odd or even or neither?
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f(x) = | x |
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g(x) = f(x+2) = | x + 2 |
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g(x) = f(x - 2) = | x - 2 |
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g(x) = f(x) + 1 = | x | + 1
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g(x) = f(x) - 1 = | x | - 1
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g(x) = 3f(x) = 3| x |
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Composition of functions
(f o g)(x) = f( g(x) )
x ----> ----> g(x) -----> ---> f(g(x)) g f
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Ex: let f(x) = 3x + 1 g(x) = x2
Find (f o g)(x)
Ex: let f(x) = 3 - x g(x) = x3
Find (g o f)(2)
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Inverse functions
Sometimes you get
x ----> ---------> ---> x
f "undid" g
g f
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If both
(f o g)(x) = x (g o f)(x) = x
Then g is the inverse of f and we write g = f-1
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Find the inverse of f(x) = 3x - 2