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Section 4.5 : Miscellaneous Functions

The point of this section is to introduce you to some other functions that don’t really require the work to graph that the ones that we’ve looked at to this point in this chapter. For most of these all that we’ll need to do is evaluate the function as some \(x\)’s and then plot the points.

Constant Function

This is probably the easiest function that we’ll ever graph and yet it is one of the functions that tend to cause problems for students.

The most general form for the constant function is,

\[f\left( x \right) = c\]

where \(c\) is some number.

Let’s take a look at \(f\left( x \right) = 4\) so we can see what the graph of constant functions look like. Probably the biggest problem students have with these functions is that there are no \(x\)’s on the right side to plug into for evaluation. However, all that means is that there is no substitution to do. In other words, no matter what \(x\) we plug into the function we will always get a value of 4 (or \(c\) in the general case) out of the function.

So, every point has a \(y\) coordinate of 4. This is exactly what defines a horizontal line. In fact, if we recall that \(f\left( x \right)\) is nothing more than a fancy way of writing \(y\) we can rewrite the function as,

\[y = 4\]

And this is exactly the equation of a horizontal line.

Here is the graph of this function.

The domain of this graph is from -5 to 5 while the range is from 0 to 6.  The graph is just a horizontal line that goes through y=4.

Square Root

Next, we want to take a look at \(f\left( x \right) = \sqrt x \). First, note that since we don’t want to get complex numbers out of a function evaluation we have to restrict the values of \(x\) that we can plug in. We can only plug in value of \(x\) in the range \(x \ge 0\). This means that our graph will only exist in this range as well.

To get the graph we’ll just plug in some values of \(x\) and then plot the points.

\(x\) \(f(x)\)
0 0
1 1
4 2
9 3

The graph is then,

The domain of this graph is from 0 to 11 while the range is from 0 to 3.5.  The graph goes through the points (0,0), (1,1), (4,2) and (9,3).  The graph increases out of the origin fairly sharply in 0<x<0.2 or so and then between 0.2<x<1 the graph bends over to the right and the rate of increase slows.  By the time the graph hits x=2 the graph looks nearly linear.  It is not a line.  There is still some curvature to the graph but it is very slight and not very clear.  The full graph is cupped vaguely downward with more curvature near the origin and only very slight curvature at the right end of the graph as already noted.

Absolute Value

We’ve dealt with this function several times already. It’s now time to graph it. First, let’s remind ourselves of the definition of the absolute value function.

\[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}x&{{\mbox{if }}x \ge 0}\\{ - x}&{{\mbox{if }}x < 0}\end{array}} \right.\]

This is a piecewise function and we’ve seen how to graph these already. All that we need to do is get some points in both ranges and plot them.

Here are some function evaluations.

\(x\) \(f(x)\)
0 0
1 1
-1 1
2 2
-2 2

Here is the graph of this function.

The domain of this graph is from -3 to 3 while the range is from 0 to 3.  The graph is basically in the shape of a “V”.  The point of the “V” is at the origin and the two arms move into the 1st and 2nd quadrant forming 45 degree angles with the x-axis.  Another way to think of the graph is that the portion that is in the 1st quadrant follows the line y=x and the portion that is in the 2nd quadrant follows the line y=-x.

So, this is a “V” shaped graph.

Cubic Function

We’re not actually going to look at a general cubic polynomial here. We’ll do some of those in the next chapter. Here we are only going to look at \(f\left( x \right) = {x^3}\). There really isn’t much to do here other than just plugging in some points and plotting.

\(x\) \(f(x)\)
0 0
1 1
-1 -1
2 8
-2 -8

Here is the graph of this function.

The domain of this graph is from -2.5 to 2.5 while the range is from -15 to 15.  The graph starts in the 3rd quadrant at (-2,-8) and increases through the point (-1,-1) and then continues to increase until it hits the origin.  It goes through the origin horizontally and then moves into the 1st quadrant and starts increasing again going through the point (1,1) and ending at that point (2,8).  The graph in the 3rd quadrant is cupped vaguely downward as it increases and the graph in the 1st quadrant is cupped vaguely upward as it increases.

We will need some of these in the next section so make sure that you can identify these when you see them and can sketch their graphs fairly quickly.