Natural Domain

Natural Domain

Domain restrictions occur when the function is undefined.

A fraction is undefined when the denominator is zero.

•  $f\left(x\right)=\frac{1}{x},\ x\ne0$  

•  $f\left(x\right)=\frac{x-9}{\left(x-2\right)\left(x+3\right)},\ x\ne-3,\ 2$  

Multiple Select

For what value(s) is the function undefined? Check ALL that apply

 $f\left(x\right)=\frac{3x-5}{x\left(2x-3\right)}$  

•  $x\ne0$  

•  $x\ne\frac{3}{2}$  

•  $x\ne\frac{5}{3}$  

• None: The Domain of this function is All Real Numbers

Multiple Select

For what value(s) is the function undefined?

 $f\left(x\right)=\frac{3x-5}{x^2-5x+6}$  

•  $x\ne3$  

•  $x\ne2$  

•  $x\ne6$  

•  $x\ne-1$  

A radical with an EVEN index must be positive or zero. It is undefined when the radicand is negative.

•  $f\left(x\right)=\sqrt{x}\ ,\ x\ge0$  

•  $f\left(x\right)=3\sqrt{x-4}+5\ ,\ x\ge4$  

• However;  $f\left(x\right)=\sqrt{x^2+3}$  has a domain of All Real Numbers because the radicand cannot be negative (no matter what we substitute for x.)

Multiple Choice

Determine the domain of the function.

 $f\left(x\right)=\sqrt{2x-3}$  

•  $x\ge\frac{3}{2}$  

•  $x>\frac{3}{2}$  

•  $x\ne\frac{3}{2}$  

•  $x\le\frac{3}{2}$  

Multiple Choice

What is the domain of the function?

 $f\left(x\right)=4\sqrt{-x+3}$  

•  $x\ge3$  

•  $x>3$  

•  $x\le3$  

• All Real Numbers

When a radical appears in the denominator of a fraction, we must consider both criteria. The radicand must be positive and CANNOT be zero.

•  $f\left(x\right)=\frac{1}{\sqrt{x}},\ x>0$  notice the  $\ge$  is not just >

•  $f\left(x\right)=\frac{3x}{\sqrt{x-2}},\ x>2$  

Multiple Choice

Determine the domain of the function.

 $f\left(x\right)=\frac{2x}{\sqrt{2x-8}}$  

•  $x>-4$  

•  $x\ne4$  

•  $x<4$  

•  $x>4$