Advanced Math / Nonlinear Functions

Nonlinear Functions and Domain Restrictions

Function questions involving radicals and rational expressions often ask for a value, an excluded input, or an endpoint of the domain. The key restrictions are that square-root radicands must be nonnegative and denominators cannot equal zero.

Least you need to know

For $\sqrt{g(x)}$, the domain requires $g(x)\ge0$.
For $\frac{p(x)}{q(x)}$, the domain excludes values where $q(x)=0$.
When a question asks for the least or greatest integer in a domain, solve the domain inequality first.
Function notation such as $f(6)$ means substitute 6 for the input.

Key notation

$f(x)=\sqrt{x-8}$ — The domain is all $x\ge8$
$r(x)=\frac{x+1}{x-6}$ — The input $x=6$ is excluded from the domain

Worked example

For $h(x)=\sqrt{5x-10}$, require $5x-10\ge0$.
Then $5x\ge10$, so $x\ge2$.
The least integer in the domain is $2$.

Common mistakes

For square-root functions, the radicand may equal zero.
For rational functions, the numerator being zero is allowed unless another restriction is given.
When the coefficient of $x$ in a domain inequality is negative, the inequality direction reverses.

How to recognize it

The problem asks for an input that is not in the domain.
The problem asks for the least or greatest integer in the domain.
The problem asks for the value of a radical or rational function at a given input.

下一節推薦課程

通過 Polynomial Function Zeros and Features 繼續學習此主題。

Polynomial Function Zeros and Features