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

2. 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

3. 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

4. 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

5. 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’

6. 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

7. 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 Now solve for x

8. 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 Now solve for x Here is the restriction on the domain of ( ƒ ◦ g )(x)

9. 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

10. 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.

11. 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) ] =

12. 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) ] =

13. 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) ] = – 2x +1 ½ – 2x < – 1

14. 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) ] = – 2x +1 ½ – 2x < – 1 Any x bigger than ½ creates a negative under the square root…

15. 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

16. 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) ?

17. 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) ?

18. 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) ? Set 6x + 24 ≥ 0 and solve for x

19. 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) ? Set 6x + 24 ≥ 0 and solve for x

20. 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) ? So (– 4) is the smallest number in the domain of (ƒ ◦ g )(x)

21. 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) ?

22. 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) ? Set and solve for x

23. 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) ? Set and solve for x These two create zero in the denominator…