Determine the inverse of 𝑓(𝑥) = 3/x-2 State the Domain and Range of the inverse

Question

Please help me with a step by step explanation

Answer 1

The inverse of f(x) = 3/(x-2) is found by replacing f(x) with 'y', flipping the variables and solving for y:

y=3 / (x-2)

Flip the variables

x = 3 / (y-2)

Multiply Both Sides by (y-2) since we need the y varible to be on its own - and it doesn't help that it's in the denominator right now

x(y-2) = 3

Still solving for y, so we're going to divide by that 'x' so y can get on its own

y-2 = 3/x

and finally add both sides by 2

y=(3/x) + 2

Domain: all possible x values. Whats important is that you cannot divide by 0! Becuase of the (3/x) term, x cannot be 0. Domain is x does not = 0

Range: all possible y values. So if X does not = 0, we need to figure out what values simply cannot exist in our inverse function. Because our inverse function is y=(3/x)+2, the inverse function will never be y=2. Why? 3/x can never be 0 (since you cannot divide by 0). Because of that, the range is y does not = 2.