# What is the domain and range of m(x) = 5/(x^2+9)??

Jul 24, 2018

Domain - all reals $x$
Range - $0 < y \le \frac{5}{9}$

#### Explanation:

Looking at the graph, you can immediately tell that it is an even function and and the graph is positive

Even function is when $f \left(x\right) = f \left(- x\right)$
$f \left(x\right) = \frac{5}{{x}^{2} + 9}$
$f \left(- x\right) = \frac{5}{{\left(- x\right)}^{2} + 9} = \frac{5}{{x}^{2} + 9} = f \left(x\right)$
Therefore, $\frac{5}{{x}^{2} + 9}$ is an even function

It is positive because ${x}^{2} + 9$ is always positive for all real integers

We also know that there is no x-intercept but there is a y-intercept at $\left(0 , \frac{5}{9}\right)$

Drawing the graph, we can see that:
Domain - all reals $x$
Range - $0 < y \le \frac{5}{9}$

For the range, $y \ne 0$ because the graph is approaching the asymptote $y = 0$ so it will never touch the line $y = 0$

The graph is below
graph{5/(x^2+9) [-10, 10, -5, 5]}