Property 1
Find the Greatest Domain of a Function
Polynomial functions are defined in all real numbers.
\(f(x)=ax^3+bx^2+cx+d\Rightarrow f:\mathbb R\to\mathbb R\)
For rational functions, denominator values that make the denominator zero are excluded.
\(f(x)=\samefrac{h(x)}{g(x)},\quad g(x)\ne0\)
Odd-index radicals are defined for every real input; even-index radicals require the radicand to be nonnegative.
\(\sqrt[2n-1]{g(x)}\): all real inputs
\(\sqrt[2n]{g(x)}\): \(g(x)\ge0\)
Exponential functions are defined on \(\mathbb R\).
\(f(x)=a^{g(x)},\ a>0\)
For logarithms, the logarithm argument must be positive.
\(f(x)=\log_a g(x),\ a>0,\ a\ne1,\ g(x)>0\)
Sine and cosine are defined on all real numbers.
\(\tan x\) is undefined at \(x=\samefrac{(2k+1)\pi}{2}\), \(k\in\mathbb Z\).
\(\cot x\) is undefined at \(x=k\pi\), \(k\in\mathbb Z\).
1.
\(f(x)=x^2+5\)
2.
\(f(x)=x^3+\sqrt5x^2-7x+4\)
3.
\(f(x)=\samefrac1{x-2}\)
4.
\(f(x)=\samefrac1{x^2-9}\)
5.
\(f(x)=\sqrt{x-4}\)
6.
\(f(x)=\sqrt[5]{x-1}+\sqrt{7-x}\)
7.
\(f(x)=\samefrac1{\sqrt{x-3}}\)