# How do you find the domain and range of y = 4x - x ^2?

Mar 1, 2017

Domain: $\left(- \infty , + \infty\right)$
Range: $\left(- \infty , 4\right]$

#### Explanation:

$y = 4 x - {x}^{2}$

$y$ is defined $\forall x \in \mathbb{R}$
Hence the domain of $y$ is $\left(- \infty , + \infty\right)$

Since the coefficient of ${x}^{2}$ is -ve, y has a maximum value where $y ' = 0$

$y ' = 4 - 2 x = 0$

$x = 2 \to {y}_{\max} = 4 \cdot 2 - {2}^{2} = 4$

Thus the range of $y$ is: $\left(- \infty , 4\right]$

This can be observed from the graph of $y$ below.

graph{x(4-x) [-6.4, 6.087, -1.347, 4.897]}