HOW TO FIND THE INVERSE OF A FUNCTION USING COMPOSITION

f(x) and g(x) are the two functions which are inverse to each other where their compositions are defined if and only if the following equations are true.

(f∘g) (x)  =  f(g(x))  =  x

 (g∘f) (x)  =  g(f(x))  =  x

Let us see some examples to understand the above concept.

Example 1 :

f(x)  =  x + 3

g(x)  =  x - 3

Are f(x) and g(x) are inverse to each other ?

Solution :

To check whether g(x) is the inverse of f(x), find f∘g(x) and g∘f(x).

Finding f∘g(x) :

f∘g(x) = f[g(x)]

f∘g(x)  =  f[x + 3]

f∘g(x)  =  x + 3 - 3

f∘g(x)  = x -----(1)

Finding g∘f(x) :

g∘f(x)  =  g[f(x)]

g∘f(x)  =  g[x - 3]

g∘f(x)  =  x - 3 + 3

g∘f(x)  =  x -----(2)

From (1) and (2), we get

f∘g(x)  =  g∘f(x)  =  x

So, f(x) and g(x) are inverse to each other. 

Example 2 :

f(x)  =  x – 3

g(x)  =  4x + 8

Are f(x) and g(x) are inverse to each other ?

Solution :

To check whether g(x) is the inverse of f(x), find f∘g(x) and g∘f(x).

f∘g(x) = f[g(x)]

f∘g(x)  =  f[4x + 8]

f∘g(x)  =  4x + 8 - 3

  f∘g(x)  = 4x + 5  ≠  x

Because f∘g(x)  ≠  x, we don't have to find g∘f(x). 

So, f(x) and g(x) are not inverse to each other. 

Example 3 :

f(x)  =  log₁₀(x)

g(x)  =  10^x

Are f(x) and g(x) are inverse to each other ?

Solution :

To check whether g(x) is the inverse of f(x), find f∘g(x) and g∘f(x).

Finding f∘g(x) :

f∘g(x) = f[g(x)]

f∘g(x)  =  f[10^x]

f∘g(x)  =  log₁₀(10^x)

f∘g(x)  =  xlog₁₀10

f∘g(x)  =  x(1) 

f∘g(x)  =  x -----(1)

Finding g∘f(x) :

g∘f(x)  =  g[f(x)]

g∘f(x)  =  g[log₁₀(x)]

g∘f(x)  =  10^log10(x)

g∘f(x)  =  x -----(2)

From (1) and (2), we get

f∘g(x)  =  g∘f(x)  =  x

So, f(x) and g(x) are inverse to each other. 

Example 4 :

f(x)  =  x – 3

g(x)  =  -5x

Are f(x) and g(x) are inverse to each other ?

Solution :

To check whether g(x) is the inverse of f(x), find f∘g(x) and g∘f(x).

f∘g(x)  =  f[g(x)]

f∘g(x)  =  f [-5x]

f∘g(x)  =  -5x - 3  ≠ x

Because f∘g(x)  ≠  x, we don't have to find g∘f(x). 

So, f(x) and g(x) are not inverse to each other. 

Example 5 :

f(x)  =  x

g(x)  =  3x + 8

Are f(x) and g(x) are inverse to each other ?

Solution :

To check whether g(x) is the inverse of f(x), find f∘g(x) and g∘f(x).

f∘g(x)  =  f[g(x)]

f∘g(x)  =  f[3x + 8]

f∘g(x)  =  3x + 8 ≠  x

Because f∘g(x)  ≠  x, we don't have to find g∘f(x). 

So, f(x) and g(x) are inverse to each other. 

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