General Mathematicsssssssss (Part 2).pdf

This is operations of functions.

1. Operations of Functions
General Mathematics

2. 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)

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

4. Subtraction of Functions
Let f and g be any two functions.
The differencef –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)

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

6. Multiplication of Functions
Let f and g be any two functions.
The product fgis 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)

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

8. Division of Functions
Let f and g be any two functions.
The quotientf/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.f
g
x()=
fx()
gx()

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

10. 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. fg fg()x()=fgx()()

11. Example 5
Given f(x) = 4x –5 and g(x) = x2+ 4,
find .
Solution to Example 5fgx() fx()=4x-5
fg()x()=4gx()()-5
=4x
2
+4( )-5
=4x
2
+11