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.

Problem 1 :

f(x) = -5x, g(x) = 19x; x = 16

Solution :

Given, f(x) = -5x and g(x) = 19x

x = 16

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = -5x + 19x

(f + g)(x) = 14x

When x = 16,

(f + g)(16) = 1416

= 14(2 2 2 2)

(f + g)(16) = 14(2)

(ii)  (f - g)(x) = f(x) - g(x)

=  -5x - 19x

(f - g)(x) = = -24x

(f - g)(16) = -2416

= -242 2 2 2

(f - g)(16) = -24(2)

(f - g)(16) = -48

Domain is set of all positive values.

Problem 2 :

f(x) = 2x, g(x) = -112x; x = -4

Solution :

Given, f(x) = 2x and  g(x) = -112x

x = -4

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x + (-112x)

(f + g)(x) = -102x

When, x = -4

(f + g)(-4) = -102(-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 - (-112x)

(f - g)(x) = 122x

(f - g)(-4) = 122(-4)

= 12(-8)

= -12(-2  -2  -2)

(f - g)(-4) = -12(-2)

(f - g)(-4) = 24

Domain is all real values.

Problem 3 :

f(x) = 6x - 4x2 – 7x3, g(x) = 9x2 – 5x; x = -1

Solution :

f(x) = 6x - 4x2 – 7x3 and g(x) = 9x2 – 5x

x = -1

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = (6x - 4x2 – 7x3) + (9x2 – 5x)

= 6x - 4x2 – 7x3 + 9x2 – 5x

= x + 5x2 – 7x3

When x = -1

(f + g)(-1) = (-1) + 5(-1)2 – 7(-1)3

(f + g)(-1) = -1 + 5 + 7

(f + g)(-1) = 11

(ii)  (f - g)(x) = f(x) - g(x)

= (6x - 4x2 – 7x3) - (9x2 – 5x)

= 6x - 4x2 – 7x3 - 9x2 + 5x

= 11x – 13x2 – 7x3

(f - g)(-1) = 11(-1) – 13(-1)2 – 7(-1)3

= -11 - 13 + 17

(f - g)(-1) = -7

Domain is all real values.

Problem 4 :

f(x) = 11x + 2x2 , g(x) = -7x – 3x2 + 4; x = 2

Solution :

f(x) = 11x + 2x2 and g(x) = -7x – 3x2 + 4

x = 2

(i)  (f + g)(x) = f(x) + g(x)

(f + g)(x) = (11x + 2x2) + (-7x – 3x2 + 4)

(f + g)(x) = 11x + 2x2 - 7x – 3x2 + 4

= 4x – x2 + 4

When, x = 2

(f + g)(2) = 4(2) – (2)2 + 4

= 8 – 4 + 4

(f + g)(2) = 8

(ii)  (f - g)(x) = f(x) - g(x)

(f - g)(x) = (11x + 2x2) - (-7x – 3x2 + 4)

= 11x + 2x2 + 7x + 3x2 - 4

(f - g)(x) = 18x + 5x2 – 4

(f - g)(2) = 18(2) + 5(2)2 - 4

= 36 + 20 - 4

(f - g)(2) = 52

Domain is all real values.

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