An inverse function is a mathematical operation that undoes the original function. In other words, if a function f(x) maps an input to an output, the inverse function will map the output back to the original input.
To find the inverse function, we first need to switch the positions of x and y in the original equation. This gives us x = y + 4/y. Next, we solve for y by multiplying both sides by y, giving us xy = y^2 + 4. Rearranging this equation to the form y^2 - xy + 4 = 0, we can then use the quadratic formula to solve for y in terms of x.
Yes, the inverse function can be written as f^-1(x) = (x + √(x^2 - 16)) / 2. This is derived from the quadratic formula solution for y in the previous step.
To check if your inverse function is correct, you can use the composition method. This means plugging the inverse function into the original function and vice versa, and making sure that the resulting equation simplifies to just x. If it does, then you have found the correct inverse function.
Yes, there are two possible inverse functions for this equation. This is because the original function is not one-to-one, meaning there are multiple inputs that can result in the same output. Therefore, there are two possible ways to map the output back to the original input, resulting in two different inverse functions.