Lesson 3

Lesson 3: Operations on Functions
Say we have two functions called f(x) and g(x). The dierent operations on
these two functions is shown below:
Adding two functions as:
(f+g)(x) = f(x)+g(x)
Subtracting two functions as:
(f-g)(x) = f(x) – g(x)
Multiplying two functions as:
(f g)(x) = f(x) g(c)
Dividing two functions as:
( f g
)(x) = f
(x ) g
(x ) ; whereg
(x ) 6
= 0
Example 1:
Let f(x) = 4x + 5 and g(x) = 3x. Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g
)(x).
(f+g)(x) = (4x+5) + (3x) = 7x+5
(f-g)(x) = (4x+5) – (3x) = x+5
(f g)(x) = (4x+5) (3x) = 12 x2
+5x
(f g
)(x) = 4
x +5 3
x
Example 2:
Let f(x)= 3x+2 and g(x)= 5x-1. Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g
)(x).
(f+g)(x) = (3x+2) + (5x-1) = 8x+1
(f-g)(x) = (3x+2) – (5x-1) = -2x+3
(f g) = (3x+2) (5x-1) = 15 x2
+7x -2
(f g
)(x) = 3
x +2 5
x 1
Example 3:
Let v(x) = x3
and w(x) = 3 x2
+5x. Find (v+w)(x), (v-w)(x), (v w)(x), and
( v w
)(x).
(v+w)(x) = ( x3
) + (3 x2
+5x) = x3
+ 3 x2
+5x
(v-w)(x) = ( x3
) (3×2
+5x) = x3
3x 2
-5x
(v w) = ( x3
) (3×2
+5x) = 3 x5
+ 5 x4
(v w
)(x) = ( x
3 3
x 2
+5 x) = x
x 2 x
(3 x+5) = x
2 3
x +5
Example 4:
Let f(x) = 4 x3
+ 2 x2
+4x + 1 and g(x) = 3 x5
+ 4 x2
+8x-12. Find (f+g)(x),
(f-g)(x), (f g)(x), and ( f g
)(x).
1

(f+g)(x) = (4 x3
+ 2 x2
+4x+1) + (3 x5
+ 4 x2
+8x-12) = 3 x5
+ 4 x3
+ 6 x2
+12x
-11
(f-g)(x) = (4 x3
+ 2 x2
+4x+1) – (3 x5
+ 4 x2
+8x-12) = 3x 5
+ 4 x3
2x 2
-4x+13
(f g)(x) = (4 x3
+ 2 x2
+4x+1) (3 x5
+ 4 x2
+8x-12)
= 12 x8
+ 6 x7
+ 12 x6
+ 19 x5
+ 40 x4
16×3
+ 12 x2
40x 12
(f g
)(x) = (4
x3
+2 x2
+4 x+1) (3
x5
+4 x2
+8 x 12)
Example 5:
Let h(x) = 1 and g(x) = x4
x3
+ x2
-1. Find (h+g)(x), (h-g)(x), (h g)(x),
and ( h g
)(x).
(h+g)(x) = (1) + ( x4
x3
+ x2
-1) = x4
x3
+ x2
(h-g)(x) = (1) – ( x4
x3
+ x2
-1) = x4
x3
+ x2
+2
(h g)(x) = (1) (x 4
x3
+ x2
-1) = x4
x3
+ x2
-1
(h g
)(x) = 1 x
4
x3
+ x2
1
2

Exercises:
1. If h(x) = 7x+3 and g(x) = 2 x2
+1. Find (f+g)(x)
2. If f(x) = x5
-18 and g(x) = x2
– 6x + 9, what is the vaue of (g-h)(x)?
3. If f(x) = 25 x5
and g(x) = 55 x8
, what is the value of ( f g
)(x)?
4. If v(x) = x3
and w(x) = x2
+ 4, what is the value of (v w)(x)?
5. If f(x) = 4x + 11 and g(x) = 5x + 9, nd (f+g)(x).
6. If f(z) = 7z – 4 and g(z) = z-2, nd (f-g)(x).
7. If f(x) =8 x2
-20 and g(x) =-4, what is the value of ( f g
)(x).
8. If f(x) = 2x+2 and g(x) = 9 x2
, what is the value of (f g)(x)?
9. If f(x) = 7 x2
+ 8x -3 and g(x) = 7x, what is the value of (f g)(x)?
10. If f(x) = 35 x8
– 45x and g(x) = 5x, what is the value of ( f g
)(x).
3

Answers to Operations on Functions Exercises:
1. 2 x2
+7x +4
2. x5
x2
+ 6x – 27
3. 5
x 11
x3
4. x5
+ 4 x3
5. 9x +20
6. 6z -2
7. 2x 2
+ 5
8. 18 x3
+ 18 x2
9. 49 x3
+ 56 x2
– 21
10.7 x7
– 9
4