1. COMPOSITION OF
FUNCTIONS
“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

2. The DOMAIN of the Composition Function
The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
The domain of g is x  1
We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x  1 so the domain of the composition would be combining the two restrictions.

3. The DOMAIN and RANGE of Composite Functions
We could first look at the natural domain and range of f(x) and g(x).
For g(x) to cope with the output from f(x) we must ensure that the output does not include 1
Hence we must exclude 6 from the domain of f(x)

4. The DOMAIN and RANGE of Composite Functions
Or we could find g o f (x) and determine the domain and range of the resulting expression.
Domain:
Range:
However this approach must be used with CAUTION.

5. The DOMAIN and RANGE of Composite Functions
We could first look at the natural domain and range of f(x) and g(x).
For f(x) to cope with the output from g(x) we must ensure that the output does not include 0
Hence we must exclude 1 from the domain of g(x)

6. The DOMAIN and RANGE of Composite Functions
Or we could find f o g (x) and determine the domain and range of the resulting expression.
Domain:
Range:
However this approach must be used with CAUTION.

7. The DOMAIN and RANGE of Composite Functions
We could first look at the natural domain and range of f(x) and g(x).

8. The DOMAIN and RANGE of Composite Functions
Or we could find g o f (x) and determine the domain and range of the resulting expression.
Not:
Domain:
Range:
However this approach must be used with CAUTION.

9. The DOMAIN and RANGE of Composite Functions
We could first look at the natural domain and range of f(x) and g(x).
f(x) can cope with all the numbers in the range of g(x) because the range of g(x) is contained within the domain of f(x)
f o g (x) is a function for the natural domain of g(x)

10. The DOMAIN and RANGE of Composite Functions
We could first look at the natural domain and range of f(x) and g(x).
g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

11. The DOMAIN and RANGE of Composite Functions
g o f (x) is not a function for the natural domain of g(x) unless we restrict the domain of f(x)
We could first look at the natural domain and range of f(x) and g(x).
g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

12. Example of Composite Functions
Try it !!

13. Method 1

14. Method 1

15. Method 1

16. Method 2

17. Method 2