Find the range of a rational function from its domain

Key concept: When a function has a restricted domain, you cannot read its range from the formula alone. You need to look at how the input interval maps through the function, then check the endpoint values carefully.

Step by step

Let

$g(x)=\frac{2x+5}{x-3}$

and suppose the domain is

$x\le 5$

Step 1: Find the boundary value

At $x=5$,

$g(5)=\frac{2(5)+5}{5-3}=\frac{15}{2}=7.5$

Step 2: Check the behaviour of the function on the domain

Rewrite the function:

$g(x)=\frac{2x+5}{x-3}=2+\frac{11}{x-3}$

For $x\le 5$, the input values move toward the asymptote at $x=3$ and also include values less than or equal to 5.

• At $x=5$, the function gives the top value $7.5$.
• As $x\to 3^{-}$, $\frac{11}{x-3}\to -\infty$, so $g(x)\to -\infty$.

That means the range on this domain is

$(-\infty,\,7.5]$

If the intended range is for a different interval, the same endpoint-and-asymptote method still works: test the boundary point, then follow the function toward any vertical asymptotes.

If your working shows a finite lower bound such as $-9$, that usually means the domain or transformation has been misread.

Pitfall alert

Don’t assume the range is just the set of values at the domain endpoints. A vertical asymptote can send the function to infinity, which completely changes the interval.

Also, be careful not to mix up the function’s original domain with the domain of its inverse or a transformed version of the function.

Try different conditions

If the domain had been $x\ge 5$, then the range would not extend to $-\infty$ through $x\to 3^{-}$, because those inputs would be excluded. In that case you would only analyze the branch for $x\ge 5$.

If a new restriction such as $5\le x\le 10$ were given, you would check both endpoints:

$g(5)=\frac{15}{2},\qquad g(10)=\frac{25}{7}$

and then determine whether the function is increasing or decreasing on that interval.

Further reading

range, vertical asymptote, restricted domain