FUNCTIONS : Domain values When combining functions using the composite rules, it is necessary to check the domain for values that could be restricted.

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…