What is the inverse of a function y1?

Answer 1

Definition of the inverse of a function.

Let f and g be two functions such that

f(g(x)) = x for every x in the domain of g and

g(f(x)) = x for every x in the domain of f.

The function g is the inverse of the function f, and the domain of f is equal to the range of g, and vice versa.

Example: Find the inverse of y1 = 2x + 7

Solution

y1 = 2x + 7 interchange x and y;

x = 2y1 + 7 solve for y;

x - 7 = 2y1 + 7 -7 subtract 7 to both sides;

x - 7 = 2y1 divide by 2 both sides;

(x - 7)/2 = y1 replace y1 with y2;

y2 = (x - 7)/2

Thus, the inverse of y1 = 2x +7 is y2 = (x -7)/2

Let's check if this is true according to the above definition:

Let y1 = f(x) = 2x +7 and y2 = g(x) = (x -7)/2

1. f(g(x))= x ?

f(x) = 2x + 7

f((x - 7)/2) = 2[(x -7)/2] + 7 = x - 7 + 7 = x True

2. g(f(x) = x ?

g(x) = (x - 7)/2

g(2x + 7) = [(2x + 7) - 7]/2 = 2x/2 = x True