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FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
To find your restrictions apply the composite rule, then :
a) set the expression in the denominator ≠ 0 and solve for x
- your domain will be all real numbers EXCEPT the restriction
b) set the expression under the root < 0 and solve for x
- your domain will the result where x ≥ OR x ≤ the restriction
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x2
1
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain of g(x) is all real numbers, but f(x) has a denominator.
So ( x – 2 ) ≠ 0
2
1
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain of g(x) is all real numbers, but f(x) has a denominator.
So ( x – 2 ) ≠ 0
Using the composite rule, replace 2x into f(x) for ‘x’
2
1
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain of g(x) is all real numbers, but f(x) has a denominator.
So ( x – 2 ) ≠ 0
Using the composite rule, replace 2x into f(x) for ‘x’
2x – 2 ≠ 0
2
1
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain of g(x) is all real numbers, but f(x) has a denominator.
So ( x – 2 ) ≠ 0
Using the composite rule, replace 2x into f(x) for ‘x’
2x – 2 ≠ 0
2
1
x
Now solve for x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain of g(x) is all real numbers, but f(x) has a denominator.
So ( x – 2 ) ≠ 0
Using the composite rule, replace 2x into f(x) for ‘x’
2x – 2 ≠ 0
2x ≠ 2
x ≠ 1
2
1
x
Now solve for x
Here is the restriction on the domain of ( ƒ ◦ g )(x)
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( ƒ ◦ g )(x) if ƒ(x) = and g (x) = 2x
The domain ( ƒ ◦ g )(x) is all Real Numbers except 1.
Because which is undefined
2
1
x
0
1
22
1)2()1(
fgf
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
The domain of f(x) is all real numbers, but g(x) is a square root.
52 x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
The domain of f(x) is all real numbers, but g(x) is a square root.
Apply the composite rule ( g ◦ ƒ )(x) = g [ f (x) ] =
52 x
532 x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
The domain of f(x) is all real numbers, but g(x) is a square root.
Apply the composite rule ( g ◦ ƒ )(x) = g [ f (x) ] =
52 x
532 x
12526532 xxx
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
The domain of f(x) is all real numbers, but g(x) is a square root.
Apply the composite rule ( g ◦ ƒ )(x) = g [ f (x) ] =
52 x
532 x
12526532 xxx
– 2x +1 < 0 x > ½
– 2x < – 1
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
The domain of f(x) is all real numbers, but g(x) is a square root.
Apply the composite rule ( g ◦ ƒ )(x) = g [ f (x) ] =
52 x
532 x
12526532 xxx
– 2x +1 < 0 x > ½
– 2x < – 1
Any x bigger than ½ creates a negative
under the square root…
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Find the domain of ( g ◦ ƒ )(x) if ƒ(x) = 3 – x, and g (x) =
Therefore the domain of ( g ◦ ƒ )(x) all real numbers where x ≤ ½
52 x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) = x + 4 . What is the smallest value in the
domain of (ƒ ◦ g )(x) ?
x6
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) = x + 4 . What is the smallest value in the
domain of (ƒ ◦ g )(x) ?
x6
246464)( xxxfxgf
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) = x + 4 . What is the smallest value in the
domain of (ƒ ◦ g )(x) ?
x6
246464)( xxxfxgf
Set 6x + 24 ≥ 0 and solve for x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) = x + 4 . What is the smallest value in the
domain of (ƒ ◦ g )(x) ?
x6
246464)( xxxfxgf
Set 6x + 24 ≥ 0 and solve for x
4
246
0246
x
x
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) = x + 4 . What is the smallest value in the
domain of (ƒ ◦ g )(x) ?
x6
246464)( xxxfxgf
So (– 4) is the smallest number in the domain of (ƒ ◦ g )(x)
4
246
0246
x
x
x
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) =
What two numbers ARE NOT in the domain of (ƒ ◦ g )(x) ?
x
1xx 42
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) =
What two numbers ARE NOT in the domain of (ƒ ◦ g )(x) ?
x
1xx 42
Set and solve for x 042 xx042 xx
FUNCTIONS : Domain values
When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.
Rules : a) you CAN NOT have a zero in the denominator
b) you CAN NOT have a negative under an even
index / root
EXAMPLE : Let ƒ(x) = and g (x) =
What two numbers ARE NOT in the domain of (ƒ ◦ g )(x) ?
x
1xx 42
Set and solve for x 042 xx
4
0
04
042
x
x
xx
xx
These two create zero in the denominator…