OPERATIONS WITH FUNCTIONS

About "Operations with Functions"

Operations with 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)  =  1/(x+4) and g(x)  =  (x + 4)³

Find the following (i) f + g  (ii)  f - g  (iii)  fg  (iv)  f/g  (v)  2f  (vi)  1/f 

Solution :

f(x)  =  1/(x + 4)

Domain of f(x)  =  R - {-4}

g(x)  =  (x + 4)³

Domain of g(x)  =  R (all real values)

There exists common domain for both the functions and it is   R - {-4}. 

(i)  f + g : R - {-4} -> R and is defined by 

(f + g) (x)  =  f(x) + g(x)

  =  [1/(x+4)] + [(x + 4)³]

  =  (1 + (x + 4)⁴)/(x + 4)

(ii)  f - g : R - {-4} -> R and is defined by 

(f - g) (x)  =  f(x) - g(x)

  =  [1/(x+4)] - [(x + 4)³]

  =  (1 - (x + 4)⁴)/(x + 4)

(iii)  f g : R - {-4} -> R and is defined by 

(f g) (x)  =  f(x) ⋅ g(x)

  =  [1/(x+4)] ⋅ [(x + 4)³]

=  (x + 4)²

(iv)  (f/g)(x)  =  R - {-4} -> R and is defined by 

g(x)  =  0

(x + 4)³  =  0

x + 4  =  0 and x  =  -4

Domain (f/g) = Domain (f) n Domain (g) - {x:g(x)=0}=R-{-4}

(f/g) (x)  =  f(x) / g(x)

  =  [1/(x+4)] / [(x + 4)³]

=  1/(x + 4)⁴

(v)  2 f 

(2f) (x)  =  2 f(x)  

  =  2 (1/(x + 4))

  =  2/(x + 4)

(vi)  1/f 

( 1/f) x  =  1/f(x)

f(x)  =  1/(x+4) 

 x + 4  =  0

x  =  -4  ≠ 0

1/f :  R - {-4} -> R and is defined by 

(1/f)(x)  =  1/[1/(x+4)]

  =  x +  4

(vii)  1/g

(1/g) x  =  1/g(x)

g(x)  =  (x + 4)³  =  0

x  =  -4  ≠ 0

1/g :  R - {-4} -> R and is defined by 

(1/g)(x)  =  1/(x+4)³

After having gone through the stuff given above, we hope that the students would have understood "Operations with Functions". 

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