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

Jun 9, 2016

Domain: $x \in \left(- \infty , \infty\right)$
Range: $f \left(x\right) \in \left(5 , \infty\right)$

#### Explanation:

The domain is easy. You can enter any real number without restriction i.e. $x \in \mathbb{R}$ or equivalently $x \in \left(- \infty , \infty\right)$.

The range is a little trickier, but you can arrive at the answer two ways. First, if you know the graph of the function, you would know that it is a parabola opening up with its vertex at (0,5). Hence $f \left(x\right) \in \left(5 , \infty\right)$.

graph{x^2+5 [-9.66, 10.34, -0.96, 9.04]}

Another way to think about the range is to consider what values $x$ can be. Because the variable is squared and a positive number is being added, the output must always be positive. Additionally, the smallest value will be achieved when $x = 0$. At this point, $f \left(x\right) = 5$ and so $f \left(x\right) \in \left(5 , \infty\right)$