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 $\,g\,$. |
 |
EXAMPLE:
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:
$\,[-1,0] \cup [1,\infty )$
|
Let's discuss the solution set of the inequality
‘$\,f(x)\gt g(x)\,$’.
Imagine a vertical line passing through the graph, moving from left to right.
Every time the vertical line is at a place where the graph of $\,f\,$ lies above the graph of $\,g\,$,
then you must include that $\,x$-value in the solution set.
Be extra careful of places where something interesting is happening
(like where the graphs intersect).
Imagine a vertical line passing through the graph, moving from left to right.
Every time the vertical line is at a place where the graph of $\,f\,$ lies above the graph of $\,g\,$,
then you must include that $\,x$-value in the solution set.
Be extra careful of places where something interesting is happening
(like where the graphs intersect).
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Composition of Functions
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Composition of Functions