SUM DIFFERENCE PRODUCT AND QUOTIENT OF FUNCTIONS

Sum Difference Product and Quotient of Functions :

Here we are going to see, how to find sum, difference product and quotient of functions.

Question 1 :

Suppose

p(x) = x² + 5x + 2, q(x) = 2x³ − 3x + 1, s(x) = 4x³ − 2

write the indicated expression as a sum of terms, each of which is a constant times a power of x.

(i)  (4p + 5q)(x)     (ii) (pq)(x)     (iii)  (ps)(x)    (iv)  (p(x))²

(v)  (q(x))²      (vi)  (p(x))²s(x)

Solution :

(i)  (4p + 5q)(x)  =  4 p(x) + 5 q(x)

  =  4(x² + 5x + 2) + 5(2x³ − 3x + 1) 

  =  4x² + 20x + 8 + 10x³ − 15x + 5

  =  10x³ +  4x²  − 15x + 20x + 5 + 8

  =  10x³ +  4x²  + 5x + 13

(ii) (pq)(x)

p(x) = x² + 5x + 2, q(x) = 2x³ − 3x + 1

(pq)(x)  =  p(x) ⋅ q(x)

  =  (x² + 5x + 2) (2x³ − 3x + 1)

  =  x² (2x³ − 3x + 1) + 5x (2x³ − 3x + 1) + 2 (2x³ − 3x + 1)

  =  2x⁵ - 3x³ + x² + 10x⁴ - 15x² + 5x + 4x³ - 6x + 2

  =  2x⁵ + 10x⁴ - 3x³ + 4x³ + x²  - 15x² + 5x - 6x + 2

  =  2x⁵ + 10x⁴ + x³ - 14x² x + 2

(iii)  (ps)(x) 

  =  p(x) ⋅ s(x)

p(x) = x² + 5x + 2  s(x) = 4x³ − 2

  =  (x² + 5x + 2) ⋅ (4x³ − 2)

  =  x²(4x³ − 2) + 5x(4x³ − 2) + 2(4x³ − 2)

  =  4x⁵ - 2x² + 20x⁴ - 10x + 8x³ - 4

  =  4x⁵ + 20x⁴ + 8x³- 2x² - 10x - 4

(iv)  (p(x))²

  =  (x² + 5x + 2)²

  =  (x²)² + (5x)² + 2² + 2 x²(5x) + 2(5x) 2 + 2 (2)x²

  =  x⁴ + 25x² + 4 + 10x³ + 20x + 4x²

  =  x⁴ + 10x³ + 29x² + 20x + 4

(v)  (q(x))²   

  =  (2x³ − 3x + 1)²

  =  (2x³)² + (-3x)² + 1² + 2 (2x³)(-3x) + 2(-3x) 1 + 2(2x³)

  =  4x⁶ + 9x² + 1 - 12x⁴ - 6x + 4x³

  =  4x⁶ - 12x⁴ + 4x³ + 9x² - 6x + 1

(vi)  (p(x))²s(x)

  =  (x⁴ + 10x³ + 29x² + 20x + 4)(4x³ − 2)

=  x⁴(4x³−2)+10x³(4x³−2)+29x²(4x³−2)+20x(4x³−2)+ 4(4x³−2)

=  4x⁷−2x⁴+40x⁶−20x³ + 116x⁵ - 58x² + 80x⁴ - 40x + 16x³ - 8

=  4x⁷+40x⁶+ 116x⁵  −2x⁴ + 80x⁴ + 16x³−20x³ - 58x² - 40x - 8

=  4x⁷+40x⁶+ 116x⁵  + 78x⁴ - 4x³ - 58x² - 40x - 8

Find

(i)  (f + g)(x)

(ii)  (f – g)(x)

and state the domain of each. Then evaluate f + g and f - g for the given value of x.

Question 2 :

f(x) = -5∜x, g(x) = 19∜x; x = 16

Solution :

Given, f(x) = -5∜x and g(x) = 19∜x

x = 16

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = -5∜x + 19∜x

(f + g)(x) = 14∜x

When x = 16,

(f + g)(16) = 14∜16

= 14∜(2 ⋅ 2 ⋅ 2 ⋅ 2)

(f + g)(16) = 14(2)

(ii)  (f - g)(x) = f(x) - g(x)

=  -5∜x - 19∜x

(f - g)(x) = = -24∜x

(f - g)(16) = -24∜16

= -24∜2 ⋅ 2 ⋅ 2 ⋅ 2

(f - g)(16) = -24(2)

(f - g)(16) = -48

Domain is set of all positive values.

Question 3 :

f(x) = ∛2x, g(x) = -11∛2x; x = -4

Solution :

Given, f(x) = ∛2x and  g(x) = -11∛2x

 x = -4

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = ∛2x + (-11∛2x)

(f + g)(x) = -10∛2x

When, x = -4

(f + g)(-4) = -10∛2(-4)

= -10∛(-8)

= 10∛(-2 ⋅ -2 ⋅ -2)

= 10(-2)

(f + g)(-4) = -20

(ii)  (f - g)(x) = f(x) - g(x)

(f - g)(x) = ∛2x - (-11∛2x)

(f - g)(x) = 12∛2x

(f - g)(-4) = 12∛2(-4)

= 12∛(-8)

= -12∛(-2 ⋅ -2 ⋅ -2)

(f - g)(-4) = -12(-2)

(f - g)(-4) = 24

Question 4 :

If f(x) = -7x + 2 and g(x) = x³ + x², find (g · f)(x).

Solution:

(g · f)(x) = g(x)·f(x)

(g · f)(x) = (x³ + x²) · (-7x + 2)

= -7x⁴ + 2x³ - 7x³ + 2x²

= -7x⁴ - 5x³ + 2x²

Question 5 :

If f(x) = 2x - 6 and g(x) = x² - 5x + 6, find f(x)/g(x).

Solution:

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