VERIFYING INVERSE FUNCTIONS BY COMPOSITION

Check the pair of functions are inverses to each other.

Example 1 :

f(x) = 4x + 3 and g(x) = (x - 3)/4

Solution :

Finding (f∘g) (x) :

(f∘g) (x) = f(g(x))

Finding (g∘f) (x) :

(g∘f) (x) = g(f(x))

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

Example 2 :

f(x) = x³ + 1 and g(x) = ∛(x-1)

Solution :

Finding (f∘g) (x) :

(f∘g) (x) = f(g(x))

Finding (g∘f) (x) :

(g∘f) (x) = g(f(x))

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

Example 3 :

f(x) = √(x - 3) and g(x) = x² + 3, x ≥ 0

Solution :

Finding (f∘g) (x) :

(f∘g) (x) = f(g(x))

Finding (g∘f) (x) :

(f∘g) (x) = f(g(x))

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

Example 4 :

f(x) = (4x + 4)/(x + 5) and g(x) = (4 - 5x)/(x - 4)

Solution :

Finding (f∘g) (x) :

(f∘g) (x) = f(g(x))

Finding (g∘f) (x) :

(g∘f) (x) = g(f(x))

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

Example 5 :

Archaeologists use models for the relationship between height and footprint length to determine the height of a person based on the lengths of the bones they discover. The relationship between height, h(x) in centimeters and foot print length x in centimeters is given by

h(x) = 1.1x + 143.6

Use this relationship to predict the footprint length for a person who is 170 cm tall.

Solution :

h(x) = 1.1x + 143.6

Let y = 1.1x + 143.6

y - 143.6 = 1.1x

x = (y - 143.6)/1.1

Replace x by f^-1(x) and y by x

f^-1(x) = (x - 143.6) / 1.1

When x = 170

f^-1(170) = (170 - 143.6) / 1.1

= 26.4/1.1

= 24 

A person who is 170 cm tall may have a footprint length of 24 cm.