Please explain how to create a table of signs

Question

Given a function: $$\begin{align} \ f(x)& = {\frac{x-1}{(9-x^2)}≥0}, \\ \end{align}$$

how would I go about setting up a table of signs for this function?

Answer 1

first you should find root of this function and then you can two optional number one of them one side of root and another,another side of root and obtain its sign. for example in your sample root is 1 and i give 0 , 2 to function f(0)<0 and f(2)>0 now that in simple root function sign change and if we have two (or even) same root sign of function not change. (sorry for my bad english)

Answer 2

The numerator of your function satisfy

$$x-1 = \left\{\begin{array}{c}\text{positive} & x>1 \\ \text{zero} & x=1 \\\text{negative} & x < 1\end{array}\right.$$

The denominator can be written $9-x^2 = (3-x)(3+x)$. Do the same things as above with the two factors $3-x$ and $3+x$ and then use this to draw the table of signs for the three factors. This will give you something like:

...............-3.......................1..............................3...............

-----------------------------0++++++++++++++++++++ $\text{ for } (x-1)$

+++++++++++++++++++++++++++++++0---------- $\text{ for } (3-x)$

-----------0+++++++++++++++++++++++++++++++ $\text{ for } (3+x)$

Now try to combine the three factors above to make the table for $f(x) = \frac{x-1}{(3-x)(3+x)}$.