A function, f, is a rule that assigns an element from one set of inputs called the domain to one (and only one) element in another set of outputs called the range. For our purposes here, think of the domain as the set of all real numbers for which the function is defined and the range as the set of all real numbers that result from applying the function. We’ll typically refer to numbers in the domain using the letter x and numbers in the range either as f(x) or y.

Tips for finding the domain and range of a function

• To find a domain, remember that we cannot divide by 0 and square roots are only defined for non-negative numbers.
• To find a range, investigate the function behaviour by evaluating the function at input values on either side of values excluded from the domain. Also, investigate the function behaviour as input values decrease (become large and negative) and increase (become large and positive).

Example 1

• [latex]f(x)=\sqrt{3(x-2)}[/latex].
• This function is only defined if [latex]x-2\ge 0[/latex], so [latex]x\ge 2[/latex]. The domain is therefore [latex]x\in [2, \infty)[/latex]. Note that square brackets indicate the interval end-point is included and curved brackets indicate the interval end-point is not included.
• This function can result in any non-negative real number, so the range is [latex]f(x)\in [0, \infty)[/latex].

Example 2

• [latex]f(x)=\frac{x}{x^2-1}[/latex].
• This function is only defined if [latex]x^2-1\neq 0[/latex], so [latex]x\neq -1[/latex] or [latex]1[/latex]. The domain is therefore any real number less than [latex]-1[/latex], or between [latex]-1[/latex] and [latex]1[/latex], or greater than [latex]1[/latex]. We can write this as [latex]x\in (-\infty, -1)\cup (-1,1)\cup (1, \infty)[/latex]. The [latex]\cup[/latex] symbol stands for “union.”
• This function can result in any real number, so the range is [latex]f(x)\in \mathbb{R}[/latex].

Example 3

• [latex]f(x)=\sqrt{\frac{x}{x-1}}[/latex].
• This function is only defined if [latex]x-1\ne 0[/latex] and [latex]\frac{x}{x-1}\ge 0[/latex]. If [latex]x\le 0[/latex], then [latex]\frac{x}{x-1}\ge 0[/latex]. Also, if [latex]x\gt 1[/latex], then [latex]\frac{x}{x-1}\gt 0[/latex]. The domain is therefore [latex]x\in (-\infty, 0]\cup (1, \infty)[/latex].
• This function can result in any non-negative real number other than [latex]1[/latex], so the range is [latex]f(x)\in [0, 1)\cup (1, \infty)[/latex].

Example 4

• [latex]f(x)=|x|-x=-2x\text{ (if }x\lt 0\text{) or }0\text{ (if }x\ge 0\text{)}[/latex].
• This function is defined for all real numbers. The domain is therefore [latex]f(x)\in \mathbb{R}[/latex].
• This function can result in any non-negative real number, so the range is [latex]f(x)\in [0, \infty)[/latex].