﻿ Graphical Interpretation of Sentences like f(x)=0 and f(x)>0
GRAPHICAL INTERPRETATIONS OF SENTENCES LIKE $\,f(x) = 0\,$ and $\,f(x) \gt 0$
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
Note:
For simplicity and brevity, I often say ‘paren’ instead of ‘parenthesis’.

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.

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

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: $\,[-6,-3]\cup [-2,0]\cup [1,3]\cup [5,6)\cup [7,9]$
Let's discuss the solution set of the inequality ‘$\,g(x)\gt 0\,$’.

Imagine a vertical line passing through the graph, moving from left to right.
Every time the vertical line touches a point that lies above the $\,x$-axis, then you must include that $\,x$-value in the solution set.
Be extra careful of ‘boundary’ or ‘transition’ points—places where something interesting is happening
(like where the graph crosses the $\,x$-axis, or where there's a break in the graph).

For example, suppose the vertical line reaches $\,x = -3\,$.
The point (the filled-in circle) is below the $\,x$-axis, so we don't want this $\,x$-value.
But then, the graph is above until we reach $\,x = -2\,$.
When $\,x = -2\,$, the point is on the $\,x$-axis, so we don't want this $\,x$-value, either.
This discussion gives the interval $\,(-3,-2)\,$ in the solution set.
Remember that the symbol ‘$\,\cup\,$’, the union symbol, is used to put sets together.
Master the ideas from this section
of Sentences like $\,f(x)=g(x)\,$ and $\,f(x)\gt g(x)$