1. **Problem Statement:** Given the function $$F(x) = \frac{x + 5}{2 - x}$$, find: i) Turning point (x, y) ii) Line of symmetry iii) Maximum or minimum value iv) Draw the graph of $$F(x)$$ v) Deduce the domain and range 2. **Important Notes:** - The function is a rational function with a vertical asymptote where the denominator is zero. - Turning points occur where the derivative $$F'(x)$$ is zero. - The line of symmetry for rational functions often relates to the vertical asymptote or the axis through the turning point. 3. **Step i) Find the turning point:** - First, find the derivative $$F'(x)$$ using the quotient rule: $$F(x) = \frac{u}{v} = \frac{x+5}{2-x}$$ $$u = x+5, \quad v = 2-x$$ $$u' = 1, \quad v' = -1$$ $$F'(x) = \frac{u'v - uv'}{v^2} = \frac{1 \cdot (2-x) - (x+5)(-1)}{(2-x)^2} = \frac{2 - x + x + 5}{(2-x)^2} = \frac{7}{(2-x)^2}$$ - Since $$F'(x) = \frac{7}{(2-x)^2} > 0$$ for all $$x \neq 2$$, the derivative never equals zero. - Therefore, there is **no turning point**. 4. **Step ii) Line of symmetry:** - The vertical asymptote is at $$x = 2$$ (where denominator is zero). - The function is undefined at $$x=2$$ and the graph has two branches separated by this vertical line. - This vertical asymptote $$x=2$$ acts as a line of symmetry for the hyperbola-like graph. 5. **Step iii) Max/Min value:** - Since $$F'(x) > 0$$ everywhere except at $$x=2$$, the function is strictly increasing on each interval separated by $$x=2$$. - There is no maximum or minimum value. 6. **Step iv) Graph of $$F(x)$$:** - Vertical asymptote at $$x=2$$. - Zero at $$x = -5$$ (numerator zero). - Horizontal asymptote as $$x \to \pm \infty$$ is found by dividing numerator and denominator by $$x$$: $$\lim_{x \to \pm \infty} F(x) = \lim_{x \to \pm \infty} \frac{1 + 5/x}{2/x - 1} = \frac{1 + 0}{0 - 1} = -1$$ - So horizontal asymptote is $$y = -1$$. 7. **Step v) Domain and range:** - Domain: All real numbers except where denominator is zero: $$\boxed{(-\infty, 2) \cup (2, \infty)}$$ - Range: Since the function approaches but never reaches $$y = -1$$ and is continuous on each interval, the range is: $$\boxed{(-\infty, -1) \cup (-1, \infty)}$$ **Final answers:** - Turning point: None - Line of symmetry: $$x = 2$$ - Max/Min value: None - Domain: $$(-\infty, 2) \cup (2, \infty)$$ - Range: $$(-\infty, -1) \cup (-1, \infty)$$