VERIFYING INVERSE FUNCTIONS USING COMPOSITION

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 g ≠ 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) = 10^x

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

Solution :

f o g :

f o g = f[g(x)]

= f[10^x]

= log₁₀(10^x)

= xlog₁₀10

= x(1) 

= x ----(1)

g o f :

g o f = g[f(x)]

= g[log₁₀(x)]

= 10^log10(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 g ≠ 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 g ≠ 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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