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So I found a question which states:

Without finding the inverse, state the domain and range of $f^{-1}$ $$f(x) = \frac{x-1}{x-4}$$ where $ x\neq 4$.

How can I find the domain and range of the function's inverse without finding its inverse in the first place?

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    $\begingroup$ What's the arrow diagram for bijective function? Make it on paper and observe it, and maybe try rotating the paper by 180 degrees. $\endgroup$
    – UmbQbify
    Commented Jun 4, 2020 at 15:30

2 Answers 2

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The domain of the inverse is the range of the original function, and vice versa (assuming the inverse exists, of course), since the inverse function is the reflection of the function over the line $y=x$.

Because the range of the function is $\mathbb R- \{1\}$ and the domain is $\mathbb R-\{4\}$, the inverse has domain $\mathbb R- \{1\}$ and range $\mathbb R-\{4\}$.

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Finding Domain and Range of Inverse Function w/out finding Inverse

In order to find the domain and range of an inverse function, firstly we have to go ahead and find the domain and range of the actual function f(x).

1. Find Dom. & Rng. of Function

Let's assume for a random function f(x) the domain is; R - {1}

Let's assume for a random function f(x) the range is; R - {4}

2. Replace Domain with Range and Range w/ Domain

random function f^-1(x) the domain is; R - {4}

random function f^-1(x) the range is; R - {1}

That's all there is to its folks! Its oversimplified but to the point.

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