Operation on Functions

1. Operations of Functions
General Mathematics

2. Lesson Objectives
At the end of the lesson, the students must be
able to:
• find the sum of functions;
• determine the difference between functions;
• identify the product of functions;
• find the quotient between functions; and
• determine the composite of a function.

3. Addition of Functions
Let f and g be any two functions.
The sum f + g is a function whose domains are
the set of all real numbers common to the
domain of f and g, and defined as follows:
(f + g)(x) = f(x) + g(x)

4. Example 1
If f (x) = 3x – 2 and g (x) = x2
+ 2x – 3,
find (f + g) (x).
Solution to Example 1
(f + g) (x) = f (x) + g (x)
= (3x – 2) + (x2
+ 2x – 3)
= x2
+ 5x – 5

5. Subtraction of Functions
Let f and g be any two functions.
The difference f – g is a function whose domains
are the set of all real numbers common to the
domain of f and g, and defined as follows:
(f – g)(x) = f(x) – g(x)

6. Example 2
Let f (x) = x2
– 5 and g (x) = 5x + 4,
find (f – g)(x).
Solution to Example 2
(f – g)(x) = f (x) – g (x)
= (x2
– 5) – (5x + 4)
= x2
– 5 – 5x – 4
= x2
– 5x – 9

7. Multiplication of Functions
Let f and g be any two functions.
The product fg is a function whose domains are
the set of all real numbers common to the
domain of f and g, and defined as follows:
(fg)(x) = f(x) · g(x)

8. Example 3
If f (x) = 3x – 2 and g (x) = x2
+ 2x – 3,
find (fg) (x).
Solution to Example 3
(fg)(x) = (3x – 2)(x2
+ 2x – 3)
= 3x (x2
+ 2x – 3) – 2(x2
+ 2x –3)
= 3x3
+ 6x2
– 9x – 2x2
– 4x + 6
= 3x3
+ 4x2
– 13x + 6

9. Division of Functions
Let f and g be any two functions.
The quotient f/g is a function whose domains
are the set of all real numbers common to the
domain of f and g, and defined as follows: ,
where g(x) ≠ 0.

10. Example 4
If f (x) = x + 3 and g (x) = x2
+ 2x – 3,
find (f/g) (x).
Solution to Example 4

11. Composition of Functions
The composition of the function f with g is
denoted by and is defined by the equation:
The domain of the composition function f  g is
the set of all x such that
1. x is in the domain of g; and
2. g(x) is in the domain of f.

12. Example 5
Given f(x) = 4x – 5 and g(x) = x2 + 4,
find .
Solution to Example 5

13. Exercise A
Determine whether or not each statement is
True or False.
1. If f(x) = x – 3 and g(x) = x + 4,
then (f – g)(x) = –7.
2. If f(x) = 4x – 12 and g(x) = x – 3,
then (f + g)(2) = –5.
3. If f(x) = x + 3 and g(x) = 4x, then (f · g)(2) = 40.
4. If f(x) = x + 6 and g(x) = 3x, then (f/g)(3) = 1.

14. Exercise B
Find f + g, f – g, fg, and f/g. If f(x) = x – 3 and
g(x) = x + 4,
1. f (x) = 3x + 4, g(x) =2x – 1
2. f (x) = 2x – 5, g(x) =4x2
3. f (x) = x – 1, g(x) = 2x2
+ x – 3
4. ,
5. ,