Domain and Range of Circles

The diagram below shows that the domain of a circle consists of all $x$x-values within the interval $a-r\le x\le a+r$a−r≤x≤a+r and the range of of a circle consists of all $y$y-values within the interval $b-r\le y\le b+r$b−r≤y≤b+r.

We can also write the domain in interval notation as $\left[a-r,a+r\right]$[a−r,a+r], and the range in interval notation as $\left[b-r,b+r\right]$[b−r,b+r].

The circle itself is not a function, but can be split into two semicircles, each of which are functions. We can rearrange the equation to make this clear:

So, for example, in the new form the circle whose centre is located at $\left(2,5\right)$(2,5) and has radius $r=3$r=3 has the equation given by $y=5\pm\sqrt{9-\left(x-2\right)^2}$y=5±√9−(x−2)2. It can be split into the two functions, say $f$f and $g$g, where $f\left(x\right)=5+\sqrt{9-\left(x-2\right)^2}$f(x)=5+√9−(x−2)2 and $g\left(x\right)=5-\sqrt{9-\left(x-2\right)^2}$g(x)=5−√9−(x−2)2.

Each of these functions are semicircles. Both functions will have the same domain; that of the original circle, given by $-1\le x\le5$−1≤x≤5. The range of $f$f becomes $5\le y\le8$5≤y≤8 and the range of $g$g becomes $2\le y\le5$2≤y≤5 as shown here.

Practice questions

Question 1

Question 2

Question 3