ADD SUBTRACT MULTIPLY AND DIVIDE FUNCTIONS

Example 1 :

What is (f + g)(x) ?

f(x) = -x + 5 and g(x) = 3x + 2

Solution :

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

= -x + 5 + 3x + 2

= 2x + 7

Example 2 :

What is (f - g)(x) ?

f(x) = 2x² + 5x - 7 and g(x) = -4x² + 2x + 3

Solution :

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

= 2x² + 5x - 7 -(-4x² + 2x + 3)

= 2x² + 5x - 7 + 4x² - 2x - 3

= 6x² + 3x - 10

Example 3 :

What is (f + g - h)(x) ?

f(x) = 4x - 7, g(x) = 3x + 18 and h(x) = -5x + 2

Solution :

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

= (4x - 7) + (3x + 18) - (-5x + 2)

= 4x - 7 + 3x + 18 + 5x - 2

Combine the like terms.

= 12x + 9

Example 4 :

What is (f ⋅ g)(x) ?

f(x) = 3x² and g(x) = 5x + 2

Solution :

(f ⋅ g)(x) = f(x)g(x)

= (3x²)(5x + 2)

= 15x³ + 6x²

Example 5 :

What is (f/g)(x) ?

f(x) = (5x + 20) and g(x)(7x + 28)

Solution :

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

= (5x + 20)/(7x + 28)

= 5(x + 4)/7(x + 4)

= 5/7 

Example 6 :

What is (f ⋅ g ⋅ h)(x) ?

f(x) = 6x - 8, g(x) = x/2 and h(x) = 4x

Solution :

(f ⋅ g ⋅ h)(x) = f(x)g(x)h(x)

= (6x - 8)(x/2)(4x)

= (6x - 8)(x)(2x)

= (6x - 8)(2x²)

= 12x³ - 16x²

Example 7 :

If f(x) = 3x + 2 and g(y) = 5y + 1, evaluate f(2)g(4).

Solution :

f(x)g(y) = (3x + 2)(5y + 1)

Substitute x = 2 and y = 4.

f(x)g(y) = (3x + 2)(5y + 1)

Substitute x =2 and y = 4

f(2)g(4) = [3(2) + 2][5(4) + 1]

= (6 + 2)(20 + 1)

= (8)(21)

= 168

Example 8 :

What is (f/g)(x) ?

f(x) = xy + 3x - 2y - 6 and g(x) = y² + y - 6

Solution :

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

= (xy + 3x - 2y - 6)/(y² + y - 6)

= [x(y + 3) - 2(y + 3)]/[(y + 3)(y - 2)]

= [(y + 3)(x - 2)]/[(y + 3)(y - 2)]

= (x - 2)/(y - 2)

Example 9 :

What is (f ⋅ g)(x) ?

f(x) = x² - 4 and g(x) = x² + 4

Solution :

(f ⋅ g)(x) = f(x)g(x)

= (x² + 4)(x² + 4)

= (x²)² - 4²

= x⁴ - 16

Example 10 :

What is (f/g)(x) ?

f(x) = ax - ay + bx - by and g(x) = ax - ay - bx + by

Solution :

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

= (ax - ay + bx - by)/(ax - ay - bx + by)

= [a(x - y) + b(x - y)]/[a(x - y) - b(x - b)]

= [(x - y)(a + b)]/[(x - y)(a - b)]

= (a + b)/(a - b)

Example 11 :

Find (f - g)(-2) when f(x) = 4x² - 2 and g(x) = x - 3.

A) 15     B) 13     C) -12     D) 19

Solution :

f(x) = 4x² - 2 and g(x) = x - 3.

To evalaute (f - g)(-2), we have to find  (f - g)(x)

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

= 4x² - 2 - (x - 3)

= 4x² - 2 - x + 3

= 4x² - x + 1

Applying x = -2

= 4(-2)² - (-2) + 1

= 4(4) + 2 + 1

= 16 + 3

= 19

(f - g)(x) = 19

So, option D is correct.

Example 12 :

Find (f/g) (-5) when f(x) = 4x - 2 and g(x) = 5x² + 14x + 2.

A) - 22/57    B) 4/57    C) 5/57    D) 5/18

Solution :

f(x) = 4x - 2 and g(x) = 5x² + 14x + 2.

To evaluate (f/g) (-5), we have to find (f/g)(x)

= f(x) / g(x)

= (4x - 2) / (5x² + 14x + 2)

It cannot be simplified further. Then, applying x = -5

(f/g) (-5) = (4(-5) - 2) / (5(-5)² + 14(-5) + 2)

= -22 / (5(25) - 70 + 2)

= -22 / (125 + 2 - 70)

= -22 / (127 - 70)

= -22 / 57

So, option A is correct.

Example 13 :

Find (fg)(2) when f(x) = x - 6 and g(x) = -3x² + 11x - 7.

A) -152     B) 24     C) -12   D) -76

Solution :

f(x) = x - 6 and g(x) = -3x² + 11x - 7.

To evaluate (fg)(2), we have to find (fg)(x) first.

f(x) g(x) = (x - 6) (-3x² + 11x - 7)

Applying x = 2, we get

f(x) g(2) = (2 - 6) (-3(2)² + 11(2) - 7)

= -4 (-12 + 22 - 7)

= -4(-19 + 22)

= -4(3)

= -12

So, option C is correct.

Example 14 :

f(x) = 4x - 7; g(x) = 8x - 5 Find f - g.

A) (f - g)(x) = 4x + 2; all real numbers

B) (f - g)(x) = -4x - 12; {x|x ≠ - 3}

C) (f - g)(x) = 12x - 12; {x|x ≠ 1}

D) (f - g)(x) = -4x - 2; all real numbers

Solution :

f(x) = 4x - 7; g(x) = 8x - 5

(f - g) (x) = (4x - 7) - (8x - 5)

= 4x - 7 - 8x + 5

= -4x - 2

So, option D is correct.

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