Date
Functions Part Two and
Page #
Arithmetic Combinations
of Functions
Learning Targets—
 I can identify the implicit and explicit form
of a function.
 I can find the domain of a function.
 I can perform arithmetic combinations on
functions.
Functions Part Two and Arithmetic Combinations of Functions
Admit Slip—
Page 69, #83 - 85
Implicit and Explicit Form of a Function
In general, when a function f is defined by
an equation in x and y, we say that the
function f is given implicitly. If it is possible to
solve the equation for y in terms of x, when
we write y = f(x) and say that the function is
given explicitly.
Implicit Form
3x+y=5
Explicit Form
y = f(x) = -3x + 5
x2 – y = 6
y = f(x) = x2 – 6
xy = 4
y = f(x) = 4/x
Important Facts About Functions
• For each x in a domain of a function f, there is
exactly one image f(x) in the range; however,
an element in the range can result from more
than one x in the domain.
• f is the symbol that we use to denote the
function. It is symbolic of the equation that
we use to get from an x in the domain to f(x)
in the range.
• If y = f(x), then x is called the independent
variable or argument of f, and y is called the
dependent variable or the value of f at x.
Finding the Domain of a Function
Often the domain of a function f is not
specified; instead, only the equation defining
the function is given. In such cases, we
agree that the domain of f is the largest set of
real numbers for which the value of f(x) is a
real number. The domain of a function f is
the same as the domain of a variable x in the
expression f(x).
Domain
The implied domain of a function is the set
of all real numbers for which the expression
is defined.
In general, the domain of a function
excludes:
• values that would cause division by zero
• values that would result in the even root
of a negative number.
Domain
Find the domain of each of the following
functions:
a) f(x) = x2 + 5x
b) f(x ) =
c) f(x ) =
3x
x
2
- 4
4 - 3 x
Domain
1. Find the domain of each function.
f(x ) = 3 x
g (x ) =
h (y ) =
2
3x
+ 5
2
+ 5
x + 4
3y - 12
Domain
2. Find the domain of each function.
4
f(x ) =
x
s(y ) =
3y
y + 5
g (x ) = 1 - 2 x
2
Domain
3. Find the domain of each function.
g (a ) =
f(t) =
g (x ) =
a - 10
3
t+ 4
1
3
x
-
x + 2
Domain
4. Find the domain of each function.
10
f(x ) =
2
x - 2x
g (x ) =
h (s) =
21
x
2
+ 5x + 6
s - 1
s - 4
Domain
When we use function in applications,
the domain may be restricted by physical or
geometric considerations.
For example, the domain of the function f
defined by f(x) = x2 is the set of all real
numbers. However, if f is used to obtain the
area of a square when the length x of a side is
known, then we must restrict the domain of f
to the positive real numbers, since the length
of a side can never be 0 or negative.
Domain
Express the area of a circle as a function of
the radius. Find the domain.
5. Express the area A of a rectangle as a
function of the length x if the length of the
rectangle is twice the width. Find the
domain.
Domain
6. Express the gross salary G of a person who
earns $10 per hour as a function of the
number x of hours worked. Find the domain.
Arithmetic Combinations
Sum
 f + g  (x ) =
Difference
 f - g  (x ) =
Product
f
Quotient
f(x ) + g (x )
f(x ) - g (x )
g  (x ) = f (x )
 f 
f(x )
  (x ) =
g (x )
g

g (x )
g (x )  0
Arithmetic Combinations
y = g (x )
y = f(x )
Find the following:
(f + g )(3 ) =
(f - g )(1 ) =
(f g ) (4 ) =
(
f
g
)(2 ) =
Arithmetic Combinations
y = g (x )
y = f(x )
7. Find the following:
(f + g )(1 ) =
(f - g )(2 ) =
(f g ) (0 ) =
(
f
g
)(3 ) =
Arithmetic Combinations
f(x ) = x
2
+ 1
(f + g )(3 ) =
(f - g )(0 ) =
(f g ) (4 ) =
(
f
g
)(5 ) =
g (x ) = x - 4
Arithmetic Combinations
f(x ) = x
2
+ 1
8. Find the following:
(f + g )(1 ) =
(f - g )(-2 ) =
(f g ) (-6 ) =
(
f
g
)(0 ) =
 2 f  (5 )=
g (x ) = x - 4
Arithmetic Combinations
f(x ) = x
2
+ 9x + 20
 f + g  (x ) =
g (x ) = x + 4
Arithmetic Combinations
f(x) = x
2
+ 9x + 20
 f - g  (x ) =
g(x) = x + 4
Arithmetic Combinations
f(x) = x
2
 f g  (x ) =
+ 9x + 20
g(x) = x + 4
Arithmetic Combinations
f(x) = x
2
 f 
  (x ) =
g
+ 9x + 20
g(x) = x + 4
Arithmetic Combinations
2
f(x ) = x
- 4x - 21
9. Find the following:
 f + g  (x )
 f - g  (x )
 f g  (x )
=
 f 
  (x ) =
g
=
=
g (x ) = x + 3
Functions Part Two and Arithmetic Combinations of Functions
Assignment—
Page 67 1-14 (Are you prepared, Concepts and Vocabulary)
Page 68 48-60 even (Find the domain)
Page 68 62, 64, 68, 70 (Use the arithmetic combinations)
Page 69 92, 96, 102 (Applications)
Functions Part Two and Arithmetic Combinations of Functions
Exit Slip—
Page 70 #104