Consider the function 𝑓(𝑥)=tan(𝑥)

Question

Consider the function

𝑓(𝑥)=tan(𝑥)

Select all intervals where the function is invertible.

[0,2𝜋]

(−5𝜋/2,−3𝜋/2)

(−∞,∞)

(𝜋/2,3𝜋/2)

[0,𝜋]

(−𝜋/2,𝜋/2)

Answer 1

tan(𝑥) is invertible on the intervals (−5𝜋/2,−3𝜋/2), (𝜋/2,3𝜋/2), and (−𝜋/2,𝜋/2) where it passes the horizontal line test which makes it a one-to-one function on these intervals.

It is not invertible on the other listed intervals. For example, on [0,2𝜋] we have tan(0) = tan(𝜋) = tan(2𝜋) = 0. So tan(𝑥) is not a one-to-one function on [0,2𝜋].