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

May 21, 2017

Domain: $\left(- \infty , + \infty\right)$
Range: $\left[5 , + \infty\right)$

#### Explanation:

$f \left(x\right) = {x}^{2} + 5$

$f \left(x\right)$ is defined $\forall x \in \mathbb{R}$

Hence, the domain of $f \left(x\right)$ is $\left(- \infty , + \infty\right)$

$f \left(x\right)$ is a parabola and hence has a single critical value.

Since the coefficient of ${x}^{2}$ is positive, $f \left(x\right)$ has an absolute minimum value and no upper bound.

Since ${x}^{2} \ge 0 \forall x \in \mathbb{R} \to f {\left(x\right)}_{\text{min}} = f \left(0\right)$

$\therefore f {\left(x\right)}_{\text{min}} = 5$

Hence, the range of $f \left(x\right)$ is $\left[5 , + \infty\right)$