**How to Add Subtract Multiply and Divide Two Functions ?**

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

The sum, difference product and quotient are defined for real functions only on their common domain. These operations do not make any sense for general functions even if their domains are same because the sum, difference, product and quotient may or may not meaningful for the elements in their common domain.

**Question 1 :**

Let f and g be two real functions defined by f(x) = √(x + 2) and g(x) = √(4 - x^{2})

(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 - x^{2})

(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 - x^{2})

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

= √(x + 2) + √(4 - x^{2})

(ii) f - g

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

= √(x + 2) - √(4 - x^{2})

(iii) fg

fg (x) = f(x) ⋅ g(x)

= √(x + 2) [√(4 - x^{2})]

= √[(x + 2) (4 - x^{2})]

= √[(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 - x^{2}) = 0

4 - x^{2} = 0

(2 - x)(2 + x) = 0

x = 2 and -2 ≠ 0

= √(x + 2) / √(4 - x^{2})]

= √(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 - x^{2}) ⋅ √(4 - x^{2})

= (4 - x^{2})

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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