VERIFYING INVERSE FUNCTIONS 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 o g = f[g(x)] = x

 g o f = 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 f(x) and g(x) are inverse to each other, find f o g and g o f.

f o g :

f o g = f[g(x)]

= f[x + 3]

= x + 3 - 3

= x ----(1)

g o f :

g o f = g[f(x)]

= g[x - 3]

= x - 3 + 3

=  x ----(2)

From (1) and (2), 

f o g = g o f = 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 :

f o g = f[g(x)]

= f[4x + 8]

= 4x + 8 - 3

= 4x + 5  ≠  x

Because f o ≠ x, we don't have to find g o f. And f(x) and g(x) are not inverse to each other. 

Example 3 :

f(x) = log10(x)

g(x) = 10x

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

Solution :

f o g :

f o g = f[g(x)]

= f[10x]

log10(10x)

= xlog1010

= x(1) 

= x ----(1)

g o f :

g o f = g[f(x)]

= g[log10(x)]

= 10log10(x)

= x ----(2)

From (1) and (2), 

f o g = g o f = 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 :

f o g = f[g(x)]

= f [-5x]

= -5x - 3  ≠ x

Because f  o ≠ x, we don't have to find g o f. And 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 :

f o g = f[g(x)]

= f[3x + 8]

= 3x + 8  x

Because f o ≠ x, we don't have to find g o f. And f(x) and g(x) are not inverse to each other. 

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