Decide if the function f(x)=(x-1)/(7x-3) may be invertible?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We can write f(x) = y = (x - 1)/(7x - 3)

If f(x) is invertible it should be possible to express x in terms of y.

y = (x - 1)/(7x - 3)

=> (7x - 3)y = (x - 1)

=> 7xy - 3y = x - 1

=> x - 7xy = 1 - 3y

=> x(1 - 7y) = (1 - 3y)

=> x = (1 - 3y)/(1 - 7y)

This shows the function is invertible and f^-1(x) = (1 - 3x)/(1 - 7x)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

A function is invertible if and only if is a bijection.

We'll check if for the given function there is an inverse function.

We'll inter-change x and y and we'll get:

x =  (y-1)/(7y-3)

Now, we'll multiply both sides by 7y-3:

x(7y - 3) = y - 1

We'll remove the brackets:

7xy - 3x = y - 1

We'll keep all the terms in y to the left side, the rest of term being moved to the right side.

7xy - y = 3x - 1

We'll factorize by y to the left side:

y(7x - 1) = 3x - 1

We'll divide by 7x - 1

y = (3x - 1)/(7x - 1)

There is an inverse function and this one is: f^-1(x) = (3x - 1)/(7x - 1)

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