Can someone help me find range and domain please ?

Question

Consider the function : f(x) = (x-1)/(x-2)

Define the range and domain of this function and write down the equations of the asymptotes .

ii) Find the a formula for the inverse of f , describing its domain and range.{I found that the inverse is 1/(x-1) + }

How to find the range and domain of these two please?

Answer 1

x=2 is a vertical asymptote

y=1 is a horizontal asymptote



The domain is all real x, x≠2

As x→∞2 (from the left) f(x)→-∞

As x→∞2 (from right) f(x)→∞

Therefore range: -∞ < f(x) < ∞, f(x)≠1 or (-∞, 1) U (1, ∞)



Let y=f(x), then

y(x-2) = (x-1), therefore

x = (2y-1) / (y-1)



f‾ ¹(x) = (2x-1) / (x-1)

The domain is the range of f(x): (-∞, 1) U (1, ∞)

The range is the domain of f(x): (-∞, 2) U (2, ∞)

Answer 2

Well, for f(x) the domain is the values that x can take and the range is the values for y as a result.



The other answer is right, but I just wanted to say one more thing - for the inverse of any function, the domain is the same as the original function's range, and the range is the same as the original function's domain. It's a clever trick that really helps if you are otherwise stuck.

Answer 3

domain : R - {2}

asymptote equation x = 2

range : R - {1}



let y = (x-1)/(x-2)

then xy - 2y = x-1

x(y-1) = 2y-1

x = (2y-1)/(y-1)

domain : R - {1}

asymptote equation y = 1

range : R - {2}