1. **State the problem:** We have the function $$y = \frac{1}{x - 5} + 2$$. We want to find: a) The value of $$x$$ for which $$y$$ is undefined. b) The value of $$y$$ for which $$x$$ is undefined. 2. **Find when $$y$$ is undefined:** The function involves a fraction $$\frac{1}{x - 5}$$. A fraction is undefined when its denominator is zero. Set the denominator equal to zero: $$x - 5 = 0$$ Solve for $$x$$: $$x = 5$$ So, $$y$$ is undefined at $$x = 5$$. 3. **Find when $$x$$ is undefined for a given $$y$$:** Rewrite the equation to solve for $$x$$: $$y = \frac{1}{x - 5} + 2$$ Subtract 2 from both sides: $$y - 2 = \frac{1}{x - 5}$$ Invert both sides: $$\frac{1}{y - 2} = x - 5$$ Solve for $$x$$: $$x = 5 + \frac{1}{y - 2}$$ For $$x$$ to be undefined, the denominator $$y - 2$$ must be zero: $$y - 2 = 0$$ Solve for $$y$$: $$y = 2$$ So, $$x$$ is undefined at $$y = 2$$. **Final answers:** a) $$x = 5$$ b) $$y = 2$$