ADDING AND SUBTRACTING FUNCTIONS

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) = -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.

Problem 2 :

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

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.

Problem 5 :

f(x) = 5 - 5x, g(x) = -3x² + 5 Find (f + g)(x).

a) -3x² - 5x + 10    b)  -8x² - 5x + 10     c)  -3x² + 5     d)  -8x + 10

Solution :

(f + g)(x) = f(x) + g(x)

= 5 - 5x - 3x² + 5

= -3x² + 5 + 5 - 5x

= -3x² - 5x + 10

So, option a is correct.

Problem 5 :

Given the function

𝑓 = {(−3, 4) (−2, 2) (−1, 0) (0, 1) (1, 3) (2, 4) (3, −1)}

and the function

𝑔 = {(−3, −2) (−2, 0) (−1, −4) (0, 0) (1, −3) (2, 1) (3, 2)}

compute the following values.

Solution :

(f + g)(x) = f(x) + g(x)

a) 

(f + g)(-3) = f(-3) + g(-3)

= 4 + (-2)

= 4 - 2

= 2

b) 

(f - g) (2) = f(2) - g(2)

= 4 + 1

= 5

c) 

fg(-1) = f(-1) g(-1)

= 0(-4)

= 0

d)

(g - f)(3) = g(3) - f(3)

= 2 - (-1)

= 2 + 1

= 3

e) 

(f/g)(-2) = f(-2) / g(-2)

= 2/0

= undefined

f) 

(g/f)(3) = g(3) / f(3)

= 2/(-1)

= -2

Problem 6 :

Use the graphs below to compute the following values. 

Solution :

a) 

(f + g) (1) = f(1) + g(1)

= 2 + 3

= 5

b) 

(f - g) (2) = f(2) - g(2)

= 3 - 3

= 0

c) 

(g - f) (3) = g(3) - f(3)

= 0 - 3

= -3

d) 

(f g) (4) = f(4) g(4)

= 0(4)

= 0

e) 

(f/g) (1) = f(1) g(1)

= 2(3)

= 6

f) 

(g/f) (4) = g(4) f(4)

= 4(0)

= 0