1. Operations on Functions

2. Combine like terms & put in descending order
The sum f + g
This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.
Combine like terms & put in descending order

3. The difference f - g Distribute negative
To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.
Distribute negative

4. Good idea to put in descending order but not required.
The product f • g
To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function.
FOIL
Good idea to put in descending order but not required.

5. Nothing more you could do here. (If you can reduce these you should).
The quotient f /g
To find the quotient of two functions, put the first one over the second.
Nothing more you could do here. (If you can reduce these you should).

6. So the first 4 operations on functions are pretty straight forward.
The rules for the domain of functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g.
For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

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

8. FOIL first and then distribute the 2
The Composition Function
This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function.
FOIL first and then distribute the 2

9. You could multiply this out but since it’s to the 3rd power we won’t
This is read “g composition f” and means to copy the g function down but where ever you see an x, substitute in the f function.
You could multiply this out but since it’s to the 3rd power we won’t

10. This is read “f composition f” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

11. 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.

12. 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)

13. 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.

14. 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)

15. 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.

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

17. 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.

18. 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)

19. 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.

20. 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.

21. A MathXTC Example of Composite Functions
Try it !!

22. Method 1

23. Method 1

24. Method 1

25. Method 2

26. Method 2

27. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar