Question

Find all the inverses associated with the following functions and state their domains. $$f(x)=2 x /(x+2)$$

Short answer

Answer: The inverse of the function $$f(x) = \frac{2x}{(x + 2)}$$ is $$f^{-1}(x) = \frac{-2x}{(x - 2)}$$. The domain of the original function is $$(-\infty, -2) \cup (-2, \infty)$$, and the domain of the inverse function is $$(-\infty, 2) \cup (2, \infty)$$.

Step-by-step solution

Rewrite the function with the notation y = f(x)

We are given the function $$f(x) = 2x / (x + 2)$$. Let's rewrite this function as $$y = 2x / (x + 2)$$.

Swap x and y to find the inverse

Now let's swap x and y to begin finding the inverse function: $$x = \frac{2y}{(y + 2)}$$.

Solve for the new y to get the inverse function

Now, we need to solve for the new y to get the inverse function: 1. Multiply both sides by (y + 2): $$x(y + 2) = 2y$$. 2. Distribute x: $$xy + 2x = 2y$$. 3. Move the terms containing y to one side: $$xy - 2y = -2x$$. 4. Factor y from the left side: $$y(x - 2) = -2x$$. 5. Divide both sides by (x - 2): $$y = \frac{-2x}{(x - 2)}$$. Now we have the inverse function, $$f^{-1}(x) = \frac{-2x}{(x - 2)}$$.

Determine the domain of the original function and its inverse

To find the domain of the original function, we need to ensure that the denominator is never equal to zero: $$x + 2 \neq 0$$. Solving for x, we find that x cannot be equal to -2. Therefore, the domain of the original function is $$(-\infty, -2) \cup (-2, \infty)$$. Similarly, for the inverse function, we ensure that the denominator is not equal to zero: $$x - 2 \neq 0$$. Solving for x, we find that x cannot be equal to 2. Therefore, the domain of the inverse function is $$(-\infty, 2) \cup (2, \infty)$$. In conclusion, the inverse of the function $$f(x) = \frac{2x}{(x + 2)}$$ is $$f^{-1}(x) = \frac{-2x}{(x - 2)}$$ with the domain of the original function being $$(-\infty, -2) \cup (-2, \infty)$$ and the domain of the inverse function being $$(-\infty, 2) \cup (2, \infty)$$.