Blog: In simple terms, how do you find the domain and range of a function?

Thursday, May 14, 2015

In simple terms, how do you find the domain and range of a function?

Hello!

Let's start from the domain. Usually this word is used in the sense "all values of $\displaystyle{x}$ which are possible to use for the given formula". There are some things which is impossible: division by zero, finding square root of the negative number, finding logarithm of non-positive number and so on.

So when we see a formula, we have to analyse whether it contains division, roots and so on. Example:

$\displaystyle{f{{\left({x}\right)}}}=\frac{\sqrt{{{x}+{2}}}}{{x}}.$

There is $\displaystyle{x}$ ...

Hello!

So when we see a formula, we have to analyse whether it contains division, roots and so on. Example:

$\displaystyle{f{{\left({x}\right)}}}=\frac{\sqrt{{{x}+{2}}}}{{x}}.$

There is $\displaystyle{x}$ in the denominator, so $\displaystyle{x}\ne{0}.$ Also, there is a square root, so $\displaystyle{x}+{2}\geq{0}$ or $\displaystyle{x}\geq-{2}.$ The resulting domain is $\displaystyle{\left[-{2},{0}\right)}\cup{\left({0},+\infty\right)}.$

Sometimes finding the domain may be difficult, but the idea is as above. Try the function $\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{2}-\sqrt{{{3}-{x}}}}}$ (isn't very difficult).

The range of a function is the set of its values. To decide whether some $\displaystyle{y}$ is in range of $\displaystyle{f{{\left({x}\right)}}}$ we have to consider the equation $\displaystyle{f{{\left({x}\right)}}}={y}$ for $\displaystyle{x}.$ If at least one solution exists, then $\displaystyle{y}$ is in the range.

For example, the range of a linear function $\displaystyle{f{{\left({x}\right)}}}={a}{x}+{b}$ is the set of all real numbers if $\displaystyle{a}\ne{0},$ and the set of the only one element $\displaystyle{\left\lbrace{b}\right\rbrace}$ if $\displaystyle{a}={0}.$ Another example: you probably know that the function $\displaystyle{g{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}$ takes all values in $\displaystyle{\left[-{1},{1}\right]}$ but no values outside; this means that the range of $\displaystyle{\sin{{\left({x}\right)}}}$ is $\displaystyle{\left[-{1},{1}\right]}.$

Finding a range may also be difficult.

Note that sometimes it is useful to restrict a domain "manually". For example, one may consider the function $\displaystyle{h}{\left({x}\right)}={{x}}^{{2}}+{1}$ on the segment $\displaystyle{\left[{1},{2}\right]}$ only. Then its domain is $\displaystyle{\left[{1},{2}\right]}$ and the range is $\displaystyle{\left[{2},{5}\right]}$ , while for the "unrestricted" function the domain is all real numbers and the range is $\displaystyle{\left[{1},+\infty\right)}.$