Download M-400 1-1 Lec Domain

Lecture Section 1 – 1
Domain
When we state the domain for a function f (x) we are listing ALL the x values that we CAN put into
the function and get a y value that is a real number.
For many functions the domain of the function is ALL REAL NUMBERS. You CAN put any real
number into the function for x and get a y value that is a real number.
We write this domain as x ∈ Reals or x ∈ ℜ
Example 1!
Example 2 !
State the domain of
State the domain of
f(x) = sin(2x)
2
f(x) = 3x − 9x
You CAN put any real number into 3x2 − 9
and get a y value that is a real number.
x ∈ Reals or x ∈ ℜ
!
You CAN put any real number into sin(2x)
and get a y value that is a real number.
x ∈ Reals or x ∈ ℜ
!
Domain Restrictions
There are functions where the domain of the function is NOT ALL REAL NUMBERS. When you put
certain real number into the function for x you DO NOT get a y value that is a real number.
Example 1!
Example 2 !
2x
x−5
You CANNOT put x = 5 into f (x)
and get a y value that is a real number.!
2(5)
10
f(5) =
=
5−5
0
which is NOT a real number.
You CANNOT put x = 3 into f (x)
! !
and get a y value that is a real number.
Example 3!
Example 4 !
f(x) =
f(x) =
f(3) = 3 - 4 = −1
which is NOT a real number.
f(x) = log(x)
f(x) = cot(x)
You CANNOT put x = 0 into f (x)
and get a y value that is a real number.
Log(0) is undefined
which is NOT a real number.
!
.
Lecture 1– 1Domain!
x-4
You CANNOT put x = 0 into f (x)
and get a y value that is a real number.
cot(0) = 1 / 0 which is undefined
which is NOT a real number.
!!
Page 1 of 5!
© 2016 Eitel
There are several common domain issues we encounter with functions in this class. The examples
below will cover the 4 most common domain issues and discuss how to state the domain for each
type of function.
Rational Functions
g(x)
A. f (x) =
f (x) has a domain restriction at all the x values where the denominator is equal
h(x)
to 0. The domain for f (x) is all real x values EXCEPT the x values where
h(x) = 0
Example 1!
f (x) =
Example 2 !
x+3
(x − 2)(x + 4)
f (x) =
x+3
(x − 5) 3 (3x − 2)
set each factor in the demominator
equal to 0 and solve for x
set each factor in the demominator
equal to 0 and solve for x
x−2=0
x=2
x−5=0
x=5
x+4=0
x = −4
Domain
{ x ∈ Reals | x ≠ 2 or − 4}
3x − 2 = 0
x = 2/3
Domain
{ x ∈ Reals | x ≠ 5 or 2 / 3}
!
!
Functions involving square roots
B. f (x) = g(x)
The square root function requires that the values under the square root must be
greater than or equal to 0. The domain for f (x) is all real x values where
g(x) ≥ 0
Example 1!
Example 2 !
f (x) = 3x − 2
set the function under the
f (x) = 4 − 5x
sign
set the function under the
sign
greater than or equal to 0 and solve for x
greater than or equal to 0 and solve for x
3x − 2 ≥ 0
3x ≥ 2
x ≥ 2/3
4 − 5x ≥ 0
−5x ≥ −4
x ≤ 4 /5
Domain
{ x ∈ Reals | x ≥ 2 / 3}
Domain
{ x ∈ Reals | x ≤ 4 / 5}
Lecture 1– 1Domain!
!
Page 2 of 5!
© 2016 Eitel
Example 3!
f (x) =
Example 4 !
x−2
x+3
set each function under a
f (x) =
sign
−5
must be greater than or equal to 0
x−2
since − 5 is negitive x − 2 must be ≥ 0
x − 2 is in the denominator
so x − 2 cannot equal to zero.
greater than or equal to 0 and solve for x
The domain is the INTERSECTION
of the two inequalities
x − 2 ≥ 0 and x + 3 ≥ 0
solve for x
x≥2
and
x ≥ −3
−5
x−2
!
x−2>0
solve for x
x>2
the INTERSECTION of the two inequalities is
x≥2
Domain
{ x ∈ Reals | x ≥ 2}
Domain
{ x ∈ Reals | x > 2}
Log Functions
C.
f (x) = log b ( g(x))
The log function requires that the values for g(x) must be greater than 0. The
domain for f (x) is all real x values where g(x) > 0
The domain is the the INTERSECTION of the two inequaltites inequalities
Example 1!
Example 2 !
f (x) = log 3 (3x − 6)
f (x) = log 3 (6 − 2x)
set the function inside the ( )
greater than 0 and solve for x
set the function inside the ( )
greater than 0 and solve for x
x−6>0
3x > 6
x>2
6 − 2x > 0
−2x > −6
x<3
Domain
{ x ∈ Reals | x > 2}
Domain
{ x ∈ Reals | x < 3}
Lecture 1– 1Domain!
!
Page 3 of 5!
© 2016 Eitel
Mixed Examples
Example 1!
f (x) =
Example 2 !
log(x − 1)
x+4
f (x) =
x +1
log(x + 3)
(x − 1) must be greater than 0
and x + 4 must be greater than 0
not equal to 0 as it is in the denominator
(x − 1) must be greater than or equal to 0
and x + 3 must be greater than 0
The domain is the INTERSECTION
of the two inequalities
x − 1 > 0 and x + 4 > 0
solve for x
x >1
and
x > −4
The domain is the INTERSECTION
of the two inequalities
x + 1 ≥ 0 and x + 4 > 0
solve for x
x ≥ −1
and
x > −3
the INTERSECTION of the two inequalities is
x >1
Domain
{ x ∈ Reals | x > 1}
!
the INTERSECTION of the two inequalities is
x ≥ −1
Domain
{ x ∈ Reals | x ≥ −1}
Example 3!
f (x) =
Example 4
3− x
log(x − 4)
f (x) =
2− x
(x − 3) (x + 4)
(3 − x) must be greater than or equal to 0
and x − 4 must be greater than 0
(2 − x) must be greater than or equal to 0
and x − 3 ≠ 0 and x − 3 ≠ 0
The domain is the INTERSECTION
of the two inequalities
3 − x ≥ 0 and x + 4 > 0
solve for x
x ≤ −1
and
x > −4
The domain is the INTERSECTION
of the two inequalities
x + 1 ≥ 0 and x + 4 > 0
solve for x
x ≥ −1
and
x > −3
the INTERSECTION of the two inequalities is
−4 < x ≤ −1
Domain
{ x ∈ Reals | − 4 < x ≤ −1}
!
the INTERSECTION of the two inequalities is
x ≥ −1
Domain
{ x ∈ Reals | x ≥ −1}
Lecture 1– 1Domain!
Page 4 of 5!
© 2016 Eitel
Trigonometric Functions
D. The domain of the 6 trigonometric functions must be examined for each function separately.
f (x) = sin(x)
f (x) = csc(x)
f (x) = cos(x)
Example 1!
f (x) = sec(x)
f (x) = tan(x)
f (x) = cot (x)
Example 2 !
f (x) = cos(x)
f (x) = sin(x)
cos(x) is defined for all Real numbers
cos(x) is defined for all Real numbers
Domain:
{ x ∈ Reals }
Domain:
{ x ∈ Reals }
Example 3!
!
! !
Example 4 !
f (x) = cot(x)
f (x) = tan(x)
cot (x)is undefined for the following values
tan(x)is undefined for the following values
!
−5π −3π −π π 3π 5π
......
,
, , , ,
.....
2
2 2 2 2 2
Domain:
π
{ x ∈ Reals | x ≠ + π • k} k ∈integers
2
Example 5!
Example 6 !
f (x) = csc (x)
f (x) = sec (x)
csc (x)is undefined for the following values
tan(x)is undefined for the following values
!
−5π −3π −π π 3π 5π
......
,
, , , ,
.....
2
2 2 2 2 2
Domain:
π
{ x ∈ Reals | x ≠ + π • k} k ∈integers
2
Lecture 1– 1Domain!
...... − 2π ,− 2π ,− π ,0, π , 2π , 3π .....
Domain:
{ x ∈ Reals | x ≠ π + π • k} k ∈integers
Page 5 of 5!
...... − 2π ,− 2π ,− π ,0, π , 2π , 3π .....
Domain:
{ x ∈ Reals | x ≠ π + π • k} k ∈integers
© 2016 Eitel