Question

to which intervals could we restrict the domain of $f$ to make it an invertible function?
choose all answers that apply:
a $-2 < x < 0$
b $-1 < x < 1$
c $0.25 < x < 0.75$

Explanation:

Step1: Recall invertibility condition

A function is invertible on a domain interval if it is one-to-one (strictly increasing or strictly decreasing) there, meaning no horizontal line intersects the function more than once on that interval.

Step2: Analyze each interval

• For interval A ($-2 < x < 0$): The function is strictly decreasing on this interval (no horizontal line intersects it twice here), so it is one-to-one.
• For interval B ($-1 < x < 1$): The function has a vertical asymptote at $x=0$, so it is not continuous here, and a horizontal line would intersect both the left and right parts of the asymptote, meaning it is not one-to-one.
• For interval C ($0.25 < x < 0.75$): The function is strictly decreasing on this interval (no horizontal line intersects it twice here), so it is one-to-one.

Answer:

A. $-2 < x < 0$
C. $0.25 < x < 0.75$