transformation problem 3: finding domain, range of inverse of a function

[khorven]

If you are solving for the domain and range of an inverse of a function, will they just be swapped? I.e. x >= 5, y >= 0, becomes x >= 0, y >= 5.

 

[Ishuda]

If you are solving for the domain and range of an inverse of a function, will they just be swapped? I.e. x >= 5, y >= 0, becomes x >= 0, y >= 5.

Normally yes, but not always if you consider them as individual functions. Consider
f₁(x) = y = \(\displaystyle \sqrt{x-5}\)
Domain is x \(\displaystyle \ge\) 5; Range is y \(\displaystyle \ge\) 0.

We would normally think that the function
f₂(x) = y = 5 + x²
had Domain x is a real number and Range y \(\displaystyle \ge\) 5 if we think of it by itself. However, it also happens to be the inverse of the first function, f₂(f₁(x))=f₁(f₂(x))=x, if we restrict x to x \(\displaystyle \ge\) 0.

 

[khorven]

I see. Thank you.