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 $\,f(x)\lt 0\,$ |
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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. |
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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. |
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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. |
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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. |
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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. |
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Pay attention to the difference between the brackets ‘$\,[\ ]\,$’ and parentheses ‘$\,(\ )\,$’ and braces ‘$\,\{\ \}\,$’ in the following solutions sets!
Why?
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.
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Graphical Interpretation
of Sentences like $\,f(x)=g(x)\,$ and $\,f(x)\gt g(x)$