If you know the graph of a function $\,f\,$,
then it is very easy to visualize the solution sets of sentences like
$\,f(x)=0\,$ and
$\,f(x)\gt 0\,$.
This section shows you how!
A key observation is that a sentence like $\,f(x) = 0\,$ or $\,f(x) \gt 0\,$
is a sentence in one variable, $\,x\,$.
To solve such a sentence, you are looking for value(s) of $\,x\,$ that make the sentence true.
The function $\,f\,$ is known, and determines the graph that you'll be investigating.
Recall that the graph of a function $\,f\,$ is a picture of all its
(input,output) pairs;
that is, it is a picture of all points of the form
$\,(x,f(x))\,$.
In particular, the $\,y$value of the point
$\,(x,f(x))\,$
is the number $\,f(x)\,$.
If $\,f(x)\gt 0\,$,
then the point
$\,(x,f(x))\,$
lies above the $\,x$axis.
If $\,f(x)=0\,$,
then the point
$\,(x,f(x))\,$
lies on the $\,x$axis.
If $\,f(x)\lt 0\,$,
then the point
$\,(x,f(x))\,$
lies below the $\,x$axis.
These concepts are illustrated below.
point $\,P(x,f(x))\,$ has $\,f(x)\gt 0\,$  point $\,P(x,f(x))\,$ has $\,f(x)=0\,$  point$\, P(x,f(x))\,$ has $\,f(x)\lt 0\,$ 
The graph of a function $\,f\,$ is shown at right. The solution set of the inequality ‘$\,f(x)\gt 0\,$’ is shown in purple. It is the set of all values of $\,x\,$ for which $\,f(x)\,$ is positive. That is, it is the set of $\,x$values that correspond to the part of the graph above the $\,x$axis. 

The graph of a function $\,f\,$ is shown at right. The solution set of the equation ‘$\,f(x)=0\,$’ is shown in purple. It is the set of all values of $\,x\,$ for which $\,f(x)\,$ equals zero. That is, it is the set of $\,x$intercepts of the graph. 

The graph of a function $\,f\,$ is shown at right. The solution set of the inequality ‘$\,f(x)\lt 0\,$’ is shown in purple. It is the set of all values of $\,x\,$ for which $\,f(x)\,$ is negative. That is, it is the set of $\,x$values that correspond to the part of the graph below the $\,x$axis. 

The graph of a function $\,f\,$ is shown at right. The solution set of the inequality ‘$\,f(x)\ge 0\,$’ is shown in purple. It is the set of all values of $\,x\,$ for which $\,f(x)\,$ is nonnegative. That is, it is the set of $\,x$values that correspond to the part of the graph that is either on or above the $\,x$axis. 

The graph of a function $\,f\,$ is shown at right. The solution set of the inequality ‘$\,f(x)\le 0\,$’ is shown in purple. It is the set of all values of $\,x\,$ for which $\,f(x)\,$ is nonpositive. That is, it is the set of $\,x$values that correspond to the part of the graph that is either on or below the $\,x$axis. 
The graph of a function $\,g\,$ with domain
$\,[6,10)\,$ is shown below.
Pay attention to the difference between the brackets ‘$\,[\ ]\,$’
and parentheses ‘$\,(\ )\,$’
and braces ‘$\,\{\ \}\,$’ in the solutions sets!

The solution set of the inequality
‘$\,g(x)\gt 0\,$’
is:
$\,(3,2)\cup (0,1)\cup (3,5)\cup [6,7)\cup (9,10)\,$
The solution set of the inequality
‘$\,g(x)\ge 0\,$’
is:
$\,(3,2]\cup (0,1]\cup[3,5]\cup[6,7]\cup[9,10)$
The solution set of the equation
‘$\,g(x)=0\,$’ is:
$\,\{2,1,3,5,7,9\}$
The solution set of the inequality
‘$\,g(x)\lt 0\,$’ is:
$\,[6,3]\cup (2,0]\cup (1,3)\cup (5,6)\cup (7,9)$
The solution set of the inequality
‘$\,g(x)\le 0\,$’ is:
$\,[6,3]\cup [2,0]\cup [1,3]\cup [5,6)\cup [7,9]$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
