Question

Suppose f(x)=eˣ for all x in the real numbers, and g(x) =ln(x²-4) for x>2. What is the inverse function g⁻¹(x) of g(x)?

Answer 1

To find the inverse function g⁻¹(x) of g(x), we need to switch the roles of x and g(x) in the equation g(x) = ln(x²-4) and solve for x. The inverse function is g⁻¹(x) = ln(x²-4).

Step-by-step explanation:

To find the inverse function g⁻¹(x) of g(x), we need to switch the roles of x and g(x) in the equation g(x) = ln(x²-4) and solve for x. Let's start:

1. Swap x and g(x) in the equation: x = ln(g(x)²-4)
2. Replace g(x) with x to obtain: x = ln(x²-4)
3. Now, solve for g⁻¹(x) by isolating x: g⁻¹(x) = ln(x²-4)

Therefore, the inverse function of g(x) = ln(x²-4) is g⁻¹(x) = ln(x²-4).