For the following two (02) items: Consider the function f(x) = 10^x - 10^ -x 10^x + 10^ -x . What is the inverse of the function?

Step-by-step solution

Let y = f(x) = 10^x - 10^ -x 10^x + 10^ -x Multiplying the numerator and the denominator by 10^x , we get: y = 10^ 2x - 1 10^ 2x + 1 Applying componendo and dividendo: 1 + y 1 - y = 10^ 2x + 1 + 10^ 2x - 1 10^ 2x + 1 - (10^ 2x - 1) 1 + y 1 - y = 2 10^ 2x 2 10^ 2x = 1 + y 1 - y Taking logarithm to the base 10 on both sides: 2x = ₁₀ ( 1 + y 1 - y ) x = 1 2 ₁₀ ( 1 + y 1 - y ) Replacing y with x to find the inverse function: f⁻¹(x) = 1 2 ₁₀ ( 1 + x 1 - x ) Answer: 1 2 ₁₀ ( 1 + x 1 - x )