What is the domain and range of y =4x - x^2?

1 Answer
Jun 15, 2018

Domain: all $x \in \left(- \infty , \infty\right)$, range: $y \in \left(- \infty , 4\right]$

Explanation:

Domain is all $x$'s that the function $y$ is not defined on, and in this case $y$ is defined for all $x$'s.

To find the range notice you can factor $y$ as $x \left(4 - x\right)$. Therefore, the roots are at $0 , 4$. By symmetry you know that the maximum will take place in the middle of that, that will say when $x = 2$. The reason its a max value is because of the negative sign on the ${x}^{2}$ term, which will make the graph a "sad smiley".

So $\max \left(y\right) = y \left(2\right) = 4 \left(2\right) - {2}^{2} = 4$

As the functions greatest value is 4 and it goes to $- \infty$ as $x \to \pm \infty$ its range is all $y \le 4$