VERIFYING IF TWO FUNCTIONS ARE INVERSE TO EACH OTHER

State if the given functions are inverse.

Problem 1 :

f(x) = (-x - 1)/(x - 2)

g(x) = (-2x + 1)/(-x - 1)

Solution :

f(x) = f(g(x))

Problem 2 :

f(x) = x + 6

g(x) = x - 6

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

= f[x - 6]

= x - 6 + 6

 = x ---> (1)

g ∘ f :

g ∘ f = g[f(x)]

= g[x + 6]

= x + 6 - 6

 = x ---> (2)

From (1) and (2),

f ∘ g = g ∘ f = x

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

Problem 3 :

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

g ∘ f :

g ∘ f = g[f(x)]

= g[5x + 2]

From (1) and (2),

f ∘ g = g ∘ f = x

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

Problem 4 :

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

g ∘ f :

g ∘ f = g[f(x)]

=g[-3x - 9]

From (1) and (2),

f ∘ g = g ∘ f = x

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

Problem 5 :

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

g ∘ f :

g ∘ f = g[f(x)]

=g[2x - 7]

From (1) and (2),

f ∘ g = g ∘ f = x

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

Problem 6 :

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

g ∘ f :

g ∘ f = g[f(x)]

=g[-4x + 8]

From (1) and (2),

f ∘ g = g ∘ f = x

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

Problem 7 :

Solution:

To check whether f(x) and g(x) are inverse to each other find f ∘ g and g ∘f

f ∘ g :

f ∘ g = f[g(x)]

= f[2x + 14]

g ∘ f :

g ∘ f = g[f(x)]

From (1) and (2),

f ∘ g = g ∘ f = x

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