Question

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

Answer 1

Domain - all reals `x`
Range - `0< y <= 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(x)=f(-x)`
`f(x)=5/(x^2+9)`
`f(-x)=5/((-x)^2+9)=5/(x^2+9)=f(x)`
Therefore, `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 `(0,5/9)`

Drawing the graph, we can see that:
Domain - all reals `x`
Range - `0< y <= 5/9`

For the range, `y !=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]}