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Lesson 3: Operations on Functions

Say we have two functions called f(x) and g(x), and the 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