Toolkit Functions

Constant Function

For the constant function $f(x)=c$ the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant $c$, so the range is the set $\{c\}$ that contains this single element. In interval notation, this is written as $[c,c]$, the interval that both begins and ends with $c$.

Domain: $(-\infty ,\infty )$

Range: $[c,c]$

Identity Function

For the identity function $f(x)=x$ there is no restriction on $x$. Both the domain and range are the set of all real numbers.

Domain: $(-\infty ,\infty )$

Range: $(-\infty ,\infty )$

Absolute Value Function

For the absolute value function $f(x)=|x|$ there is no restriction on x. The outputs are $x$ for $x\geq 0$ and $-x$ for $x<0$, so the range is all numbers greater than or equal to 0.

Domain: $(-\infty ,\infty )$

Range : $[0,\infty )$

Quadratic Function

For the quadratic function $f(x)=x^{2}$ the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

Domain: $(-\infty ,\infty )$

Range : $[0,\infty )$

Cubic Function

For the cubic function $f(x)=x^{3}$ the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

Domain: $(-\infty ,\infty )$

Range: $(-\infty ,\infty )$

Rational Function

For the rational function (also known as the reciprocal function) $f(x)={\frac {1}{x}}$ we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0.

Domain: $(-\infty ,0)\cup (0,\infty )$

Range: $(-\infty ,0)\cup (0,\infty )$

Squared Rational Function

For the squared rational function $f(x)={\frac {1}{x^{2}}}$ we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. Also, since $x^{2}>0$ for all $x\neq 0$, the range will only consist of positive numbers.

Domain: $(-\infty ,0)\cup (0,\infty )$

Range: $(0,\infty )$

Square Root Function

For the square root function $f(x)={\sqrt {x}}$ we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number $x$ is defined to be positive, even though the square of the negative number $-{\sqrt {x}}$ also gives us $x$.

Domain: $[0,\infty )$

Range: $[0,\infty )$

Cube Root Function

For the cube root function $f(x)={\sqrt[{3}]{x}}$ the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).

Domain: $(-\infty ,\infty )$

Range: $(-\infty ,\infty )$

Resources

• Domain Range and Toolkit Functions, Book Chapter
• Guided Notes
• Toolkit Functions, Lumen Learning