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.
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