Functions and Graphs Lesson #1: Review of Function Notation and Operations
Functions and Graphs
Page 1
Review of Function Notation and Operations
A function is defined as a relation that “maps” one element from the input set into ________
element from the output set.
For example:
A.
-2
B.
4
1
1
3
9
-2
3
5
6
7
A is a function because it maps –2 to ___, 1 to ____, and 3 to ____
B is not a function because it maps 3 to ____ and ____
Function Notation
Function notation is a method of writing equations ( y  3x  4 , y  x 2  4 x  3 ) as specific
functions ( f x  3x  4 , g x   x 2  4 x  3 ). This rearranges our speech to discuss “what
the function is doing” over an interval rather than focussing on little numbers in a table of
values.
Remember:
f(x) refers to the height of a function at a specific x value.
In function notation:
 f(x) means “f at x” or “f of x” - the height of f at x.
 f(x) tells us that the “name” of the function is f
 If f(-2) = 3, then the x (horizontal) value of the function is ____, and the y
or f (vertical) value of the function is ______.
Consider the following functions: f x  3x  4 , g x   x 2  4 x  3 , to determine:
2
a) g(1)
b) g 3
c) f  
d) g 1 2x
3
 
The graph of a function is shown. Find the following, then write as coordinates:
a) f(2) =
b) f(-1)
(___ , ___)
c) f(x)=-3
(___ , ___)
(___ , ___)
(___ , ___)
Functions and Graphs Lesson #1: Review of Function Notation and Operations
Page 2
Function Operations
The following properties apply to functions f and g:
The sum of f and g
The difference of f and g
The product of f and g



The quotient of f and g

 f  g x  f x  gx
 f  g x  f x  gx
 fg x  f x gx
f
f x 
 x  
, g x   0
g x 
g
ALWAYS REWRITE IT ! Don’t be a hero….:
Consider the functions f x   7  2 x , g x   4 x  1 and hx   2 x 2  9 x  7 to determine the
following:
a)
f
 
 g  2
b)  g  h  5
h 
d)  x 
g 
c)  fhx 

Use the graph to find the following:
** state restrictions *
Functions and Graphs Lesson #1: Review of Function Notation and Operations
a)  f  g 0
b)  fg  2
Page 3
g
c)  3
f
Function Composition
A function can be “broken up” in to two or more functions. This is called a “Composite
Function”.
A composite function composed of two functions f and g is given the formula:  f  g x  .
This is said to be the f of g of x. In other words, the g(x) function is inside the f(x) function.
This can be written in the following property:
Composition Property:
f
 g x  f g x
Remember:
Use the functions f x   x 2  1 and g x  4  3x to determine the following:
a)
g  f x
b)
f
-
SAY IT OUT LOUD !!
REWRITE IT
Plug in
Solve from the INSIDE out.
 g x 
Use the functions f x   x 2  1 and g x  4  3x to determine the following:\
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Functions and Graphs Lesson #1: Review of Function Notation and Operations
a) g  f  3
-
 
b)  f  g  5
SAY IT OUT LOUD !!
REWRITE IT
Plug in
Solve from the INSIDE out.
Assignment:
1. Evaluate the following given the functions f x   1  2 x  3x 2 , g  x   9  x ,
3
h x  
, and ix   7  3x
x4
2
a) i 
b) g  7
c) f 5
3
 
d) ha 
e) h3x  8
f) f x  1
2. Evaluate the following given the functions f  x   1  x , g x   3x 2  18 , and
hx  4 x  2 :
1
a) g  
b) h2  f  3
c) f 2 x  x 2
 3

d) hx  2a 
g) g  2  2 f  3  h 1
e) g x  3  hx  1
2
f)

h x  b   h x 
b
   
h) f  7   h 2  g 3
3. Given the graph, find the following:
g(x)
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Functions and Graphs Lesson #1: Review of Function Notation and Operations
a) i) f(-1)=
ii) f(3)=
iii) f(x)=2
b) i) g(-2)=
ii) g(0)=
iii) g(x)=0
c) i) (f+g)(0)
ii) (f – g)(1)
iii)  f  g  2
f(x)
4. If f x   x  5 , g x   2 x  5 , hx   x 2  9 , and ix   x  3 , Determine:
a)  f  g x
5. If f x  
a)  fg 3
b) i  hx
c) gf x
h 
d)  x  *state restriction*
i 

x4
, and g x   3x 2  x , find:
2x
b) g  f 4
6. Use the functions f x   2 x 2  3x , g x  
the following:
a.  f  g  9
c)  f  g 1
2
1
x  6 , h x   2 x  8 , and i  x   to find
3
x
b. g  i 2
c. i  h17
7. If f x   2 x 2  3x  1 , g x  4x  7 , and h x  
1
, find the following:
x
b. g  g x 
a.  f  g x
c. g  f x
Answer Key:
1a) 5
1b) 4
1c) 16  2 5
1d)
3
a4
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Functions and Graphs Lesson #1: Review of Function Notation and Operations
1e)
1
x4
55
3
2e) 3x 2  14 x  39
2a)
3a) 4, 0, x = 1

4a) 3 5
4d) x  3, x  3
5a) –4
1f) 3x 2  8 x  6
2b) 20
2c) 1 – x or x – 1
2d) 4 x  8a  2
2f) 4
2g) 30
2h) 6 2  25

3b) 2, -3, x = -3, -1, 2
4b)  x 2  x  12 or  x  4x  3
5b) 44
5c)
17
3
6a) 324
6b) 
7a) 32 x 2  124 x  120
7b) 16 x  35
3c) 0, 6, 10

4c) 2 x  5 x  25
1
2
6c)
1
6
7c) 8 x 2  12 x  3