The range of a function

Functions: Functions

The range of a function

The values which a function #f:\blue X\to \orange Y# takes lie within the codomain #\orange{Y}#. However, not every element from the codomain #\orange{Y}# has to occur as a function value of #f#, as will be apparent in the following example.

The function #f: \blue{\mathbb{R}} \to \orange{\mathbb{R}}# with function rule #f(x)=x^2# has the minimum value #f(0)=0# and may further take all values greater than #0#.

Thus, all of the values in the interval #\ivco{0}{\infty}# are reached by #f#. We therefore call #\ivco{0}{\infty}# the range of #f#.

In the graph it can be seen that the horizontal line #y=\orange a# has points of intersection with the graph only if #\orange a# falls within the range of #f#.

We now give a formal definition of the range of a function #f#.

Range

For a function #f: \blue X \to \orange Y#, #\orange Y# is called the codomain.

The set of all function values #f(x)# in #\orange Y# for #x \in \blue X# is called the range of #f#.

If the range is equal to #\orange Y#, we call #f# surjective.

example

Of #f: \blue{\mathbb{R}} \to \orange{\mathbb{R}}#
with function rule #f(x)=-x^2#

the range is #\ivoc{-\infty}{0}#

Determining point in range

Let #f:\blue{X} \to \orange{\mathbb{R}}# be a real-valued function and #p# a real number in #\blue{X}# with #f(p)=0#. #p# is then called a zero of #f#. Equivalently, #0# is an element of the range. Conversely, #f# has a zero if #0# lies in the range. So in order to check if #0# is an element of the range of the equation, we can check if #f(x) = 0# has a solution.

More generally, an element #y# is within the range of #f# if and only if the equation #f(x) = y# has a solution. This provides a method to determine the range.

Example

The constant function #f: \blue{\{1,2\}} \to \orange{\{1,2\}}# with #f(1)=1# and #f(2)=1# is not surjective, because the element #2# from the codomain is not within the range.

In contrast, the function #f: \blue{\{1,2\}} \to \orange{\{1,2\}}# with #f(1)=2# and #f(2)=1# is surjective, because all elements of the codomain are within the range.

Onto

The word surjective comes from the French word "sur". The translation of this is "onto" (as in "on top of"). Therefore, the phrase "a surjective function #\blue X# to #\orange Y# " is sometimes replaced by "a function of #\blue X# on #\orange Y# ".

The range of the function #f(x) = -{{4}\over{x+6}}# consists of all real numbers except one. Which number is not within the range?

The number #0# is not within the range.

A point #y# is within the range of #f# if and only if the equation #y = -{{4}\over{x+6}}# has a solution in #x#
For #y \neq 0#,
\[\begin{array}{rcl}
y &=&\displaystyle -{{4}\over{x+6}}\\
&&\phantom{xxx}\blue{\text{given function}}\\
y(x+6)&=& -4\\
&&\phantom{xxx}\blue{\text{multiplied both sides by }x+6}\\
x+6 &=& \dfrac{-4}{y}\\
&&\phantom{xxx}\blue{\text{divided by }y \text{ (note that }y\ne 0\text{)}} \\
x &=&\displaystyle \frac{-4}{y}-6\\
&&\phantom{xxx}\blue{\text{added } -6 \text{ on both sides}}
\end{array}\]

For #y=0#,
\[ \begin{array}{rcl}
0 &=&\displaystyle -{{4}\over{x+6}}\\
&&\phantom{xxx}\blue{y=0\text{ entered in function}}\\
0(x+6)&=& -4\\
&&\phantom{xxx}\blue{\text{multiplied both sides by }x+6}\\
0&=& -4\\
&&\phantom{xxx}\blue{\text{calculated}}
\end{array}\]

We conclude:

For each #y\ne0#, the equation #y = -{{4}\over{x+6}}# has a solution #x#, namely #x=-{{4}\over{y}}-6#.
For #y=0#, the equation becomes #0= -{{4}\over{x+6}}#; it does not have a real solution.

Therefore, #0# is the only real number that is not within the range of #f(x)=-{{4}\over{x+6}}#.

New example