Find the inverse

Question

Find the inverse of 𝑦=ln(𝑥²−3) for 𝑥>3^1/2.

(Use symbolic notation and fractions where needed.)

𝑓^-1(𝑥)=?

Answer 1

Write x = ln(y² – 3) and solve for y.

Take the exponential of both sides to get

e^x = y² – 3

Add 3 to both sides.

y² = e^x + 3

Take the positive square root of both sides.

y = √(e^x + 3)

answer

𝑓^–1(𝑥) = √(e^x + 3)

The negative square root is also a proper inverse function.

Note that y = ±√(e^x + 3) is NOT a proper inverse function, however. It fails the vertical line test.