HOW TO ADD SUBTRACT MULTIPLY AND DIVIDE TWO FUNCTIONS

About "How to Add Subtract Multiply and Divide Two Functions"

How to Add Subtract Multiply and Divide Two Functions ?

Here we are going to see how to add, subtract, multiply and divide functions.

Operations with Functions - Examples

Question 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 :

f(x)  =   √(x + 2) 

√(x + 2)  ≥ 0

By applying the value greater than or equal to -2, we will get positive values.

So, the domain is [-2, ∞)

and g(x)  =  √(4 - x²)

(2 - x) (2 + x)  ≥ 0

By applying the values lesser than -2, we will get negative values. By applying  the values greater than -2 upto 2, we will get positive values and equal to 0.

Domain is [-2, 2]

So, common domain for both the function is [-2, 2]

(i) f + g 

f(x)  =  √(x + 2) and g(x)  =  √(4 - x²)

(f + g) (x)  =  f(x) + g(x)

  =  √(x + 2) + √(4 - x²)

(ii)  f - g 

(f - g) (x)  =  f(x) - g(x)

  =  √(x + 2) - √(4 - x²)

(iii)  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)(x)  =  f(x)/g(x)

g(x)  =  √(4 - x²)  =  0

4 - x²  =  0

(2 - x)(2 + x)  =  0

x  =  2 and -2 ≠ 0

  =   √(x + 2) / √(4 - x²)]

  =   √(x + 2) / √(2 + x) (2 - x)

 f(x)/g(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)

After having gone through the stuff given above, we hope that the students would have understood "How to Add Subtract Multiply and Divide Two Functions". 

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