Graphical Interpretation of Sentences like f(x)=g(x) and f(x)>g(x)

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.

The notation

$\,P(x,f(x))\,$ is a convenient shorthand for: the point $\,P\,$ with coordinates $\,(x,f(x))$

$P_1(x,f(x))\,$ 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 following solutions sets!

The solution set of the inequality ‘$\,f(x)\gt g(x)\,$’ is:

$\,(-\infty ,-1) \cup (0,1)$
Why?
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).

Remember that the symbol ‘$\,\cup\,$’, the union symbol, is used to put sets together.

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:

$\,[-1,0] \cup [1,\infty )$

You can use GeoGebra to explore the ideas from this section (link below).
(Please be patient. It may take a few minutes for GeoGebra to load. It's worth the wait!)

GeoGebra Worksheet: Graphical Interpretation of Sentences like $\,f(x) = g(x)\,$ and $\,f(x) \gt g(x)$

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

(MAX is 15; there are 15 different problem types.)