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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)
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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