# How do you know if a function is nonlinear?

May 24, 2016

When there it has any exponents, logarithm or root involving the variable x. These always result in a curve.

#### Explanation:

A linear function is defined as:

$f \left(x\right) = a \cdot x + b$

Note that is has two parameter - the $a$, which multiplies the variable x, and the constant $b$. There are no powers or roots in this function. Adding any kind of power to this expression will make it turn into a curve. Some examples of nonlinear functions:

$y = {x}^{2} + 1$.

$y = {e}^{x} + 1$.

$y = \ln \left(x\right)$, where $x > 0$.

$y = \frac{1}{x}$, which is the same as $y = {x}^{- 1}$ and $x \ne 0$.

$y = \sqrt{x}$, which is the same as $y = {x}^{\frac{1}{2}}$ and $x \ge 0$.

Therefore, if we add any power to $x$ or place it in a logarithm or under a division, the result will be a curve. Also, the function cannot be linear if it has a restriction at its domain or range.