How do you find the domain and range of f(x)=(x^2+5x-6)/(x^2-5x+6)?

Jul 20, 2017

The domain is $\mathbb{R} - \left\{2 , 3\right\}$
The range is $\left(- \infty , - 49\right] \cup \left[- 1 , + \infty\right)$

Explanation:

For the domain, the denominator must be $\ne 0$

Therefore,

The domain is $\mathbb{R} - \left\{2 , 3\right\}$

For the range, let rewrite the function

$y = \frac{{x}^{2} + 5 x - 6}{{x}^{2} - 5 x + 6}$

$y \left({x}^{2} - 5 x + 6\right) = {x}^{2} + 5 x - 6$

$y {x}^{2} - {x}^{2} - 5 y x - 5 x + 6 y + 6 = 0$

$\left(y - 1\right) {x}^{2} - 5 \left(y + 1\right) x + 6 \left(y + 1\right) = 0$....................$\left(1\right)$

Let 's calculate the discriminant of equation $\left(1\right)$

$\Delta \ge 0$

$25 {\left(y + 1\right)}^{2} - 24 \left(y - 1\right) \left(y + 1\right) \ge 0$

$25 \left({y}^{2} + 2 y + 1\right) - 24 \left({y}^{2} - 1\right) \ge 0$

${y}^{2} + 50 y + 49 \ge 0$

$\left(y + 49\right) \left(y + 1\right) \ge 0$

The range is $\left(- \infty , - 49\right] \cup \left[- 1 , + \infty\right)$

graph{(x^2+5x-6)/(x^2-5x+6) [-132.6, 134.3, -94.2, 39.3]}