Rational Function 187428

1. **Problem statement:** We are given the rational function $$y = \frac{5 + x}{2 - x}$$ and its equivalent form $$y = \frac{x + 5}{-x + 2}$$. 2. **Goal:** Understand why these two forms are equivalent and analyze the function. 3. **Step 1: Recognize the equivalence of numerator and denominator terms.** - The numerator $$5 + x$$ is the same as $$x + 5$$ because addition is commutative. - The denominator $$2 - x$$ can be rewritten as $$-(x - 2)$$ or $$-x + 2$$. 4. **Step 2: Rewrite the function using the denominator transformation:** $$y = \frac{x + 5}{2 - x} = \frac{x + 5}{-(x - 2)} = -\frac{x + 5}{x - 2}$$ 5. **Step 3: Compare with the given transformed form:** The given transformed form is $$y = \frac{x + 5}{-x + 2}$$ which is the same as $$y = -\frac{x + 5}{x - 2}$$. 6. **Step 4: Conclusion:** The two forms are equivalent because the denominator differs by a factor of $$-1$$, which can be factored out, changing the sign of the entire fraction. 7. **Additional notes:** - The function has a vertical asymptote at $$x = 2$$ because the denominator is zero there. - The function is undefined at $$x = 2$$. **Final answer:** The two forms $$y = \frac{5 + x}{2 - x}$$ and $$y = \frac{x + 5}{-x + 2}$$ represent the same rational function, differing only by a factor of $$-1$$ in the denominator which changes the sign of the fraction.