# How do you find the domain and range for y=2x-10?

May 5, 2016

The domain and range are both $\mathbb{R}$ (all real numbers).

#### Explanation:

The domain consists of all possible valid $x$-values, while the range consists of all possible $y$-values.

As $2 x - 10$ is defined for all real numbers, the domain of $2 x - 10$ is $\mathbb{R}$.

We would have to limit our domain if there was the possibility of dividing by $0$ such as in $\frac{1}{x}$ for $x = 0$, taking an even root of a negative number, such as $\sqrt{x}$ for $x < 0$, or taking the logarithm of a non-positive number, such as $\ln \left(x\right)$ for $x \le 0$. As none of these apply, we do not have any restrictions on our choice of $x$.

Note that $2 \left(\frac{y + 10}{2}\right) - 10 = y$, and thus for any real $y$, we may choose $x = \frac{y + 10}{2}$ to get $y = 2 x - 10$. As such, every real number $y$ is in the range of the function.