The document outlines key objectives for a mathematics lesson on operations with functions, including the sum, difference, product, quotient, and composition of functions. It provides definitions and examples for each operation, demonstrating how to perform these operations with given functions. Additionally, it includes exercises to test comprehension of these concepts through true/false statements and function evaluations.

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. ,