# How do you find the domain and range of y= 5sqrt((-x+3)^4)?

Nov 4, 2017

Donain: $\left(- \infty , + \infty\right)$ Range: $\left[0 , + \infty\right)$

#### Explanation:

$y = 5 \sqrt{{\left(- x + 3\right)}^{4}}$

$= 5 \times {\left(- x + 3\right)}^{\frac{4}{2}} = 5 {\left(- x + 3\right)}^{2}$

$y$ is defined $\forall x \in \mathbb{R}$

Hence, the domain of $y$ is: $\left(- \infty , + \infty\right)$

${y}_{\min} = y \left(3\right) = 0$

$y$ has no finite uper bound.

Hence, the range of $y$ is: $\left[0 , + \infty\right)$

As can be deduced from the graph of $y$ below.

graph{5sqrt((-x+3)^4) [-2.27, 7.593, -0.676, 4.254]}