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:

A function is invertible if it is one-to-one (passes the horizontal line test) on its domain. This means the function must be strictly increasing or strictly decreasing over the interval, with no repeated y-values.

• For interval A ($-2 < x < 0$): The function is strictly decreasing across this entire interval, so it passes the horizontal line test.
• For interval B ($-1 < x < 1$): This interval crosses a vertical asymptote, so the function is not continuous here, and it fails the horizontal line test (it increases on one side of the asymptote and decreases on the other).
• For interval C ($0.25 < x < 0.75$): The function is strictly decreasing across this entire interval, so it passes the horizontal line test.

Answer:

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