Function

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Function

• A FUNCTION is a mathematical rule that for each
  input (x-value) there is one and only one
  output (y value).
• Set of Ordered Pairs
• (input, output) or (x , y)
• No x-value is repeated!!!
• A function has a DOMAIN (input or x-values) and a
  RANGE (output or y-values)
• For Graphs, Vertical Line Test
• If a vertical line can be drawn anywhere on the
  graph that it touches two points, then the graph
  is not a function

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Function Representations f is 2 times a number
plus 5
Set of Ordered Pairs (-4, -3), (-2, 1), (0, 5),
(1, 7), (2, 9)
Mapping
Table
x y
-4 -3
-2 1
0 5
1 7
2 9
Graph
-4 -2 0 1 -2
-3 1 5 7 9
Function Notation
f(-4) -3 f(1) 7 f(-2) 1 f(2) 9 f(0) 5
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Examples of a Function
1 Graphs
4 Mapping
2 Table
3 Set
x y
-1 -8
3 4
4 7
6 13
7 16
(2,3), (4,6), (7,8), (-1,2), (0,4), (-2, 5),
(-3, -2)
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Non Examples of a Function
1 Graphs
2 Table
4 Mapping
x -1 2 1 0 -1
y -5 3 2 -1 4
3 Set
(-1,2), (1,3), (-3, -1), (1, 4), (-4, -2), (2,
0)
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Practice Is it a Function?
1 No

• (2,3), (-2,4), (3,5), (-1,-1), (2, -5)
• (1,4), (-1,3), (5, 3), (-2,4), (3, 5)
• 3. 5.
• 4. 6.

2 Yes
5 No
3 No
x 3 2 3 5 -1
y -2 4 8 -1 2
6 Yes
4 No
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Function Notation

• Function Notation just lets us see what the
  INPUT value is for a function. (Substitution
  Statement)
• It also names the function for us most of the
  time we use f, g, or h.
• Examples f(x) 2x
• Reads as f of x is 2 times x
• f(3) 2 (3) 6
• The (3) replaces every x in rule for the input.
• Examples g(x) 3x2 7x
• Reads as g of x is 3 times x squared minus 7
  times x
• g(-1) 3(-1)2 7(-1) 10
• The (-1) replaces every x in rule for the input.

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Given f a number multiplied by 3 minus 5 f(x)
3x 5
2) Find f(2)
3) Find f(3x)
1) Find f(-4)
3( -4) 5 -12 5 -17
3( 2) 5 6 5 1
3( 3x) 5 9x 5
5) Find f(x) f(2)
4) Find f(x 2)
3( x 2) 5 3x 6 5 3x 1
3(x) 5 3(2) 5 3x 5 1 3x
4
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Given g a number squared plus 6g(x) x2 6
1) Find g(4)
2) Find g(-1)
3) Find g(2a)
( 4)2 6 16 6 22
( -1)2 6 1 6 7
( 2a)2 6 4a2 6
5) Find g(x - 1)
4) Find 2g(a)
( x-1)2 6 x2 2x 1 6 x2 2x 7
2( a)2 6 2a2 12
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Operations on Functions
Operations Notation
Sum
Difference
Product
Quotient
Example 1 Add / Subtract Functions
a)
b)
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Example 2 Multiply Functions
a)
b)
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Composite Function
Combining a function within another function.
Notation
Function f of Function g of x
x to function g and then g(x) into function f
Example 1 Evaluate Composites of Functions
Recall (a b)2 a2 2ab b2
a)
b)
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Example 2 Composites of a Function Set
a)
g(x)
f(x)
f(g(x))
X Y
7 3
5 3
9 8
11 4
X Y
3 5
5 7
7 9
9 11
X Y
3 3
5 3
7 8
9 4
This means f(g(5))3
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Example 2 Composites of a Function Set
b)
g(x)
f(x)
In set form, not every x-value of a composite
function is defined
X Y
7 3
5 3
9 8
11 4
X Y
5 7
3 5
7 9
9 11
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Evaluate Composition Functions

• Find
• f(g(3)) b) g(f(-1)) c) f(g(-4))
• d) e) f)

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Inverse Functions and Relations
Inverse Relation
Relation (function) where you switch the domain
and range values
Function ? Inverse
Function ? Inverse
Domain of the function ? Range of Inverse and
Range of Function ? Domain of Inverse
Inverse Notation
Input a into function and output b, then inverse
function will input b and output a (switch)
Composition of function and inverse or vice versa
will always equal x (original input)
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Steps to Find Inverses
1 Replace f(x) with y
2 Interchange x and y
One-to-One
A function whose inverse is also a function
(horizontal line test)
Function
Inverse
Inverse is not a function
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Example 1 One-to-One (Horizontal Line Test)
Determine whether the functions are one-to-one.
a)
b)
One-to-One
Not One-to-One
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Example 2 Inverses of Ordered Pair Relations
a)
Are inverses f-1(x) or g-1(x) functions?
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Inverses of Graphed Relations

• FACT The graphs of inverses are reflections
  about the line y x

Find inverse of y 3x - 2
y 1/3x 2/3
x 3y 2 x 2 3y 1/3x 2/3 y
y x
y 3x - 2
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Example 3 Find an Inverse Function
a)
b)
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Example 3 Continued
c)
d)
PART D) Function is not a 1-1. (see example) So
the inverse is 2 different functions If you
restrict the domain in the original function,
then the inverse will become a function. (x gt 0
or x lt 0)
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Example 4 Verify two Functions are Inverses

1. Method 1 Directly solve for inverse and check

• Method 2 Composition Property

Yes, Inverses
Yes, Inverses