HOW TO ADD SUBTRACT MULTIPLY AND DIVIDE TWO FUNCTIONS

Example 1 :

Let f and g be two real functions defined by

f(x) = √(x + 2) and g(x) = √(4 - x²)

(i) f + g  (ii)  f - g  (iii)  fg  (iv)  f/g  (v)  ff  (vi)  gg.

Solution :

(i) f + g

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

= f(x) + g(x)

= √(x + 2) + √(4 - x²)

(ii) f - g

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

= f(x) - g(x)

= √[(x + 2) - √(4 - x²)]

(iii) fg

fg= fg(x)

= f(x)g(x)

= √(x + 2)√(4 - x²)

= √[(x + 2)(4 - x²)]

= √[(x + 2)(2 - x)(2 + x)]

  = √[(x + 2)²(2 - x)]

= (x + 2)√(2 - x)

(iv) f/g

f/g = (f/g)(x)

= f(x)/g(x)

= √(x + 2)/√(4 - x²)]

= √(x + 2)/√[(2 + x) (2 - x)]

 = 1/√(2 - x)

(v) ff

ff(x) = f(x)f(x)

= √(x + 2)√(x + 2)

= (x + 2)

(vi) gg

gg(x) = g(x)g(x)

= √(4 - x²)√(4 - x²)

= (4 - x²)

Example 2 :

Given

f(x) = x – 3 ; g(x) = 2x² – 5x – 3 ; h(x) = 4x ; j(x) = 4x² + 12x

find each function.

a) (f + g)(x)

b)  (g + j)(x)

c)  (h + f)(7)

d)  (j - g)(x)

e)  (h - f)(-3)

f) (f • j)(-1)

h) (f • g) (x)

i) (g ∘ f)(x)

Solution :

f(x) = x – 3 ; g(x) = 2x² – 5x – 3 ; h(x) = 4x ; j(x) = 4x² + 12x

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

= x - 3 + 2x² – 5x – 3

= 2x² – 5x + x - 3 – 3

= 2x² - 4x- 6

b)  (g + j)(x) = g(x) + j(x)

= 2x² – 5x – 3 + 4x² + 12x

= 2x² + 4x² – 5x + 12x – 3

= 6x² + 7x – 3

c)  (h + f)(7) = h(7) + f(7)

(h + f)(x) = h(x) + f(x)

= 4x + x - 3

(h + f)(x) = 5x - 3

Applying the value of x, we get

(h + f)(7) = 5(7) - 3

= 35 - 3

= 32

d)  (j - g)(x) = j(x) - g(x)

= 4x² + 12x - (2x² – 5x – 3)

= 4x² + 12x - 2x² + 5x + 3

= 4x² - 2x² + 12x + 5x + 3

= 2x² + 17x + 3

e)  (h - f)(-3) 

(h - f)(x) = h(x) - f(x)

= 4x - (x - 3)

= 4x - x + 3

(h - f)(x) = 3x + 3

(h - f)(-3) = 3(-3) + 3

= -9 + 3

= -6

f) (f • j)(-1)

(f • j)(x) = f(x) • j(x)

=  (x - 3) (4x² + 12x)

= 4x³ + 12x² - 12x² - 36x

(f • j)(x) = 4x³ - 36x

(f • j)(x) = 4(-1)³ - 36(-1)

= 4(-1) + 36

= -4 + 36

= 32

h) (f • g) (x) = f(x) • g(x)

= (x - 3) (2x² – 5x – 3)

i) (g ∘ f)(x) = g[f(x)]

= g[x -3]

applying x - 3 in the place of x in the function g(x), we get

= 2(x - 3)² – 5(x - 3) – 3

= 2(x² - 6x + 9) - 5x + 15 - 3

= 2x² - 11x + 18 - 5x + 15 - 3

= 2x² - 16x + 15

Example 3 :

Which of the following represents (g • f) (5),

if f(x) = -13-x² and g(x) = -14 ?

a)  168     b)  532     c)  494    d) 196

Solution :

To find (g • f) (5), we have to evaluate (g • f) (x).

Given that, f(x) = -13-x² and g(x) = -14

(g • f) (5) = g(5) f(5)

Evalauting g(5), we get 

g(5) = -14

Evalauting f(5), we get 

f(5) = -13 - 5²

= -13 - 25

= -38

g(5) f(5) = -14(-38)

= 532

So, the answer is 532.

Example 4 :

A function f satisfies f(2) = 3 and f(3) = 5.A function g satisfies g(3) = 2 and g(5) = 6. What is the value of f(g (3)) ?

a) 2    b) 3   c) 5    d) 6

Solution :

f(g (3)) = f[g(3)]

Applying the value of g(3), we get

= f[2]

Applying the value of f(2), we gtet

= 3

So, the answer is option b.

g(x) = ax² + 24

For the following g defined above, a is constant and g(-4) = 8. What is the value of g(-4) ?

a) 8      b)  0      c)  -1     d) -8

g(4) = 8

First let us apply a = 4, we get

g(4) = a(4)² + 24

8 = 16a + 24

8 - 24 = 16a 

16a = -16

a = -1

Applying the value of a, we get

g(x) = -x² + 24

g(-4) = -(-4)² + 24

= -16 + 24

= 8

So, the answer is option a.

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