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. |
![]() |