Let f : R → R be defined as f ( x ) = 10 x + 7. Find the function g : R → R such that g o f = f o g = 1 R .
The given function is f ( x ) = 10 x + 7 defined in f : R → R .
The inverse of the function is to be determined. To find the inverse, first check whether the function is one-one and onto.
For one-one:
Let f ( x ) = f ( y ) [ x , y ∈ R ] Put the values of the functions.
10 x + 7 = 10 y + 7 x = y Thus, the function is one-one.
For onto:
For y ∈ R , let the function be defined as y = 10 x + 7 .
y − 7 10 = x For all values of y belonging to R, the value of x also belongs to R such that f ( x ) = y .
f ( x ) = f ( y − 7 10 ) = 10 ( y − 7 10 ) + 7 = y Thus, the function is onto.
Since the function is one-one and onto, the function is invertible.
Now, we can define the inverse function of f as,
g ( y ) = y − 7 10 The value of g ∘ f ( x ) is,
g ( f ( x ) ) = g ( 10 x + 7 ) = 10 x + 7 − 7 10 = 10 x 10 = x
Similarly, the value of f ∘ g ( y ) is,
f ( g ( y ) ) = f ( y − 7 10 ) = 10 ( y − 7 10 ) + 7 = y
Thus, g ∘ f ( x ) = f ∘ g ( y ) = I R .
Thus, the required function is g ( y ) = y − 7 10 .
The given function is
The inverse of the function is to be determined. To find the inverse, first check whether the function is one-one and onto.
For one-one:
Let
For onto:
For
Since the function is one-one and onto, the function is invertible.
Now, we can define the inverse function of f as,
Similarly, the value of
Thus,
Thus, the required function is
