Describe in words, as accurately as possible, the domain and range of the functions illustrated here.

Graph of square root three x times one minus x which has maximum height 2/3 and intersects the x-axis at 1
Graph of one over x minus 1, which has a vertical asymptote at x=1 and crosses the y-axis at -1

Can you write the domains and ranges using mathematical notation?

The function on the left is undefined for any negative value of \(x\) because of the square root. It is defined at \(x=0\), so we could describe its domain as any real number greater than or equal to zero. \(f(0)=0\) and as \(x\) increases \(f(x)\) increases up to \(\frac{2}{3}\) then decreases. The function’s range is any real number less than or equal to \(\frac{2}{3}\). \[\text{domain: }x\in\mathbb{R}\text{, }x\ge0\quad\text{ range: }f(x)\in\mathbb{R}\text{, }f(x)\le\frac{2}{3}\]

The symbols \(\in\mathbb{R}\) are mathematical notation meaning “is a member of the set of all real numbers”.

The function on the right is defined for all real values of \(x\) except for \(x=1\). Its range is all real numbers greater than zero and all real numbers less than zero. Notice that it is impossible to obtain the output value \(f(x)=0\). \[\text{domain: }x\in\mathbb{R}\text{, }x\ne1\quad\text{ range: }f(x)\in\mathbb{R}\text{, }f(x)\ne0\]

Sometimes, the domain and range might be written in an abbreviated form such as \[\text{domain: }x\ne1\quad\text{ range: }f(x)\ne0 \text{ .}\]

We refer to real numbers as opposed to imaginary or complex numbers. You may learn about those elsewhere.