1. **State the problem:** We have the function $$f : x \mapsto \frac{2x}{x-6}$$ with $$x \neq 6$$. (a) Find $$f(10)$$. (b) Find the inverse function $$f^{-1}$$ in the form $$f^{-1} : x \mapsto ...$$. 2. **Find $$f(10)$$:** Substitute $$x=10$$ into the function: $$f(10) = \frac{2 \times 10}{10 - 6} = \frac{20}{4} = 5$$. 3. **Find the inverse function $$f^{-1}$$:** Start with the equation: $$y = \frac{2x}{x-6}$$ We want to express $$x$$ in terms of $$y$$. Multiply both sides by $$x-6$$: $$y(x-6) = 2x$$ Distribute $$y$$: $$yx - 6y = 2x$$ Bring all $$x$$ terms to one side: $$yx - 2x = 6y$$ Factor out $$x$$: $$x(y - 2) = 6y$$ Divide both sides by $$y - 2$$: $$x = \frac{6y}{y - 2}$$ Show cancellation if any (none here), so this is the inverse function. 4. **Write the inverse function:** $$f^{-1} : x \mapsto \frac{6x}{x - 2}$$ **Final answers:** (a) $$f(10) = 5$$ (b) $$f^{-1}(x) = \frac{6x}{x - 2}$$