HOW TO FIND COMPOSITION OF TWO FUNCTIONS WITH ABSOLUTE VALUE

Example 1 :

Let f, g : R → R be defined as f(x) = 2x − |x| and g(x) = 2x + |x|. Find f ◦ g and g ◦ f

Solution :

We know that,

 |x|  =  x  if x > 0 or 

 |x|  =  -x  if x < 0 

Now let us find f(x) and g(x) respectively.

Given : f(x) = 2x − |x|

f(x)  =  2x - x if x > 0

 f(x)  =  x    if  x > 0

f(x)  =  2x - (-x)  if x < 0

  =  2x + x 

 f (x)   =  3x   if x < 0

Given : g(x) = 2x + |x|

g(x)  =  2x + x if x > 0

 g(x)  =  3x    if  x > 0

g(x)  =  2x - x  if x < 0

  =  2x - x 

=  x

g (x)   =  x   if x < 0

Now we have to find the composition of the above functions.

f ◦ g (x)  when x > 0

f ◦ g (x)  =  f [g (x)] 

  =  f [ 3x ]

we have to apply the value 3x instead of x in the function f (x)

 f ◦ g (x)  =  3x

g ◦ f (x)  when x < 0

g ◦ f (x)  =  g [f (x)] 

  =  g [ 3x ]

we have to apply the value 3x instead of x in the function g (x)

 g ◦ f (x)  =  3x

Example 2 :

If f, g : R → R are defined by f(x) = |x| + x and g(x) = |x| − x, find g ◦ f and f ◦ g.

Solution :

We know that,

 |x|  =  x  if x > 0 or 

 |x|  =  -x  if x < 0 

Now let us find f(x) and g(x) respectively.

Given : f(x) = |x| + x

f(x)  =  x + x if x > 0

 f(x)  =  2x    if  x > 0

f(x)  =  x - x  if x < 0

  =  0   if x < 0

Given : g(x) = |x| - x 

g(x)  =  x - x if x > 0

 g(x)  =  0    if  x > 0

g(x)  =  -x - x  if x < 0

  =  -2x 

g (x)   =  -2x   if x < 0

Now we have to find the composition of the above functions.

f ◦ g (x)  when x > 0

f ◦ g (x)  =  f [g (x)] 

  =  f [ 0 ]

we have to apply the value 0 instead of x in the function f (x)

f(0)  =  2 (0)  =  0

Hence f ◦ g (x)  =  0

g ◦ f (x)  when x < 0

g ◦ f (x)  =  g [ f (x)] 

  =  g [ 0 ]

we have to apply the value 0 instead of x in the function f (x)

f(0)  =  -2 (0)  =  0

g ◦ f (x)  =  0

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