How do you find the domain and range of f(x)=x?

Aug 14, 2017

The domain of $f \left(x\right) = x$ is the whole of the real numbers $\mathbb{R}$. The range is also the whole of $\mathbb{R}$.

Explanation:

Given:

$f \left(x\right) = x$

• The domain of $f \left(x\right)$ is the set of values for which $f \left(x\right)$ is defined. In the context of Algebra I that means a subset of the real numbers $\mathbb{R}$. In the case of the given $f \left(x\right)$, it is well defined for any $x \in \mathbb{R}$, so the domain is the whole of $\mathbb{R}$, i.e. $\left(- \infty , \infty\right)$

• The range of $f \left(x\right)$ is the set of values that it can take for some value of $x$. Given any real number $y$, let $x = y$. Then $f \left(x\right) = x = y$. So the range of $f \left(x\right)$ is the whole of $\mathbb{R}$ too.

The graph of $f \left(x\right) = x$ is a diagonal line like this:

# graph{x [-10, 10, -5, 5]}

For every $x$ coordinate there is a corresponding point on the line. That tells us that the domain of $f \left(x\right)$ is the whole of $\mathbb{R}$.

For every $y$ coordinate there is a corresponding point on the line. That tells us that the range of $f \left(x\right)$ is the whole of $\mathbb{R}$.