This section should feel remarkably similar to the previous one: graphical interpretation of sentences like $\,f(x) = 0\,$ and $\,f(x) \gt 0\,$. This current section is more general—to return to the previous ideas, just let $\,g(x)\,$ be the zero function.
If you know the graphs of two functions $\,f\,$ and $\,g\,$,
then it is very easy to visualize the solution sets of sentences like
$\,f(x) = g(x)\,$ and
$\,f(x)\gt g(x)\,$.
This section shows you how!
A key observation is that a sentence like
$\,f(x) = g(x)\,$ or $\,f(x) \gt g(x)\,$
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 functions $\,f\,$ and $\,g\,$ are known, and determine the graphs that you'll be investigating.
Recall that the graph of a function $\,f\,$ is a picture of all points of the form
$\,(x,f(x))\,$,
and the graph of a function $\,g\,$ is a picture of all points of the form
$\,(x,g(x))\,$.
In particular, the $\,y$value of the point
$\,(x,f(x))\,$
is the number $\,f(x)\,$
and the $\,y$value of the point
$\,(x,g(x))\,$ is the number $\,g(x)\,$.
If $\,f(x)\gt g(x)\,$,
then the point
$\,(x,f(x))\,$
lies above the point $\,(x,g(x))\,$.
If $\,f(x)=g(x)\,$,
then the graphs of $\,f\,$ and $\,g\,$ intersect at this point.
If $\,f(x)\lt g(x)\,$,
then the point
$\,(x,f(x))\,$ lies below the point $\,(x,g(x))\,$.
These concepts are illustrated below.
$P_1(x,f(x))\,$
and
$\,P_2(x,g(x))\,$
with $\,f(x)\gt g(x)\,$ 
$P_1(x,f(x))\,$
and
$\,P_2(x,g(x))\,$
with $\,f(x)=g(x)$ 
$P_1(x,f(x))\,$
and
$P_2(x,g(x))\,$
with $\,f(x)\lt g(x)$ 
for this value of $\,x\,$, the graph of $\,f\,$ lies above the graph of $\,g$ 
for this value of $\,x\,$, the graphs of $\,f\,$ and $\,g\,$ intersect 
for this value of $\,x\,$, the graph of $\,f\,$ lies below the graph of $\,g$ 
The graphs of functions
$\,f\,$ and $\,g\,$ are shown at right.
The solution set of the inequality ‘$\,f(x)\gt g(x)\,$’ is shown in green. It is the set of all values of $\,x\,$ for which the graph of $\,f\,$ lies above the graph of $\,g\,$. 

The graphs of functions
$\,f\,$and $\,g\,$ are shown at right.
The solution set of the equation ‘$\,f(x)=g(x)\,$’ is shown in green. It is the set of all values of $\,x\,$ for which the graphs of $\,f\,$ and $\,g\,$ intersect. 

The graphs of functions
$\,f\,$ and $\,g\,$ are shown at right.
The solution set of the inequality ‘$\,f(x)\lt g(x)\,$’ is shown in green. It is the set of all values of $\,x\,$ for which the graph of $\,f\,$ lies below the graph of $\,g\,$. 

The graphs of functions
$\,f\,$ and $\,g\,$ are shown at right.
The solution set of the inequality ‘$\,f(x)\ge g(x)\,$’ is shown in green. It is the set of all values of $\,x\,$ for which the graph of $\,f\,$ lies on or above the graph of $\,g\,$. 

The graphs of functions
$\,f\,$and $\,g\,$ are shown at right.
The solution set of the inequality ‘$\,f(x)\le g(x)\,$’ is shown in green. It is the set of all values of $\,x\,$ for which the graph of $\,f\,$ lies on or below the graph of $\,g\,$. 
The graphs of two functions, each with domain $\,\mathbb{R}\,$, are shown below:
$\,f\,$ is a parabola, shown in the black dotted pattern;
$\,g\,$ is a cubic polynomial, shown in purple.
These two curves intersect at the points
$\,(1,2)\,$,
$\,(0,0.5)\,$
and $\,(1,1)\,$.
Pay attention to the difference
between the brackets ‘$\,[\ ]\,$’
and parentheses ‘$\,(\ )\,$’
and braces ‘$\,\{\ \}\,$’
in the solutions sets!
Remember that the symbol
‘$\,\cup\,$’,
the union symbol,
is used to put sets together.

The solution set of the inequality
‘$\,f(x)\gt g(x)\,$’
is:
$\,(\infty ,1) \cup (0,1)$
The solution set of the equation
‘$\,f(x)=g(x)\,$’
is:
$\,\{1,0,1\}\,$
The solution set of the inequality
‘$\,f(x)\lt g(x)\,$’
is:
$\,(1,0)\cup (1,\infty)\,$
The solution set of the inequality
‘$\,f(x)\ge g(x)\,$’
is:
$\,(\infty ,1] \cup [0,1]$
The solution set of the inequality
‘$\,f(x)\le g(x)\,$’
is:
$\,[1,0] \cup [1,\infty )$

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
