# What is the domain and range of f(x)= (2-x)/(x^2+7x+12)?

Jun 29, 2017

Domain: $x \in \mathbb{R}$ except $x = - 3 \mathmr{and} x = - 4$. In interval notation:$\left(- \infty , - 4\right) \cup \left(- 4 , - 3\right) \cup \left(- 3 , \infty\right)$ , Range : Any real number i.e $f \left(x\right) \in \mathbb{R}$. In interval notation: $\left(- \infty , \infty\right)$

#### Explanation:

$f \left(x\right) = \frac{2 - x}{{x}^{2} + 7 x + 12} \mathmr{and} f \left(x\right) = \frac{2 - x}{\left(x + 4\right) \left(x + 3\right)}$

Domain: Input restriction is denominator should not be $0$

So $\left(x + 4\right) \ne 0 \mathmr{and} x \ne - 4 \mathmr{and} \left(x + 3\right) \ne 0 \mathmr{and} x \ne - 3$

Domain: $x \in \mathbb{R}$ except $x = - 3 \mathmr{and} x = - 4$. In interval notation:

$\left(- \infty , - 4\right) \cup \left(- 4 , - 3\right) \cup \left(- 3 , \infty\right)$

Range : Any real number i.e $f \left(x\right) \in \mathbb{R}$. In interval notation:

$\left(- \infty , \infty\right)$

graph{(2-x)/(x^2+7x+12) [-160, 160, -80, 80]} [Ans]