Finding the range of rational functions

Question

I have a problem that I cannot figure out how to do. The problem is:
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?

I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have tried to find the inverse of function s but I got stuck trying to isolate y. Here is what I have done so far:
$y=\frac{x+2}{x^2+5}$

$x=\frac{y+2}{y^2+5}$

$x(y^2+5)=y+2$
$xy^2+5x=y+2$
$xy^2-y=2-5x$
$y(xy-1)=2-5x$

This is the step I got stuck on, usually I would just divide by the parenthesis to isolate y but since y is squared, I cannot do that. Is this the right approach to finding the range of the function? If not how would I approach this problem?

Answer 1

To find the range, we want to find all $y$ for which there exists an $x$ such that $$ y = \frac{x+2}{x^2+5}.$$ We can solve this equation for $x$: $$ y x^2 + 5y = x+2$$ $$ 0 = y x^2 -x + 5y-2$$ If $y \neq 0$, this is a quadratic equation in $x$, so we can solve it with the quadratic formula: $$ x = \frac{ 1 \pm \sqrt{ 1 - 4y(5y-2)}}{2y}.$$ So, for a given $y$, $y$ is in the range if this expression yields a real number. That is, if $$ 1 - 4y(5y-2) = -20y^2 +8y +1 \ge 0$$ If you study this quadratic, you will find that it has roots at $y=1/2$ and $y=-1/10$, and between these roots it is positive, while outside these roots it is negative. Hence, there exists an $x$ such that $s(x)=y$ only if $$ -\frac{1}{10} \le y \le \frac{1}{2}. $$ Thus, this is the range of $s$.

(Note we excluded $y=0$ earlier, but we know $y=0$ is in our range since $s(-2)=0$.)

Answer 2

Hint: one way would be to sketch it, notice the minimum and maximum (call them $m$ and $M$), and show that $y>M$ and $y<m$ lead to contradictions (show also that $y=M$ and $y=m$ for certain values of $x$)

Answer 3

Going back to your original problem statement, you have to understand that the range of a function is the set of values that the function takes on for arguments in the function's domain. This avoids worrying about functions that are not 1-1. (I don't see the need for the "domain of $s^{-1}$" statement.) I assume that the domain in your case is the real numbers.

Once you understand this, then you can apply the variety of analyses that show you how to get the range for your particular function.