In this section we study the graphs of some important basic functions. Many functions fall into families or classes of similar functions, and recognizing the appropriate family for a given situation is an important part of modeling.
We can use the calculator to find cube roots as follows. Press the MATH key to get a menu of options. Option 4 is labeled \(\sqrt[3]{~~}\text{;}\) this is the cube root key. To find the cube root of, say, 15.625, we key in \(\qquad\qquad\)MATH4\(15.625\))ENTER and the calculator returns the result, 2.5. Thus, \(\sqrt[3]{15.625} = 2.5\text{.}\) You can check this result by verifying that \(~2.5^3=15.625\text{.}\)
We use the absolute value to discuss problems involving distance. For example, consider the number line below. Starting at the origin, we travel in opposite directions to reach the two numbers \(6\) and \(-6\text{,}\) but the distance we travel in each case is the same.
The distance from a number \(c\) to the origin is called the absolute value of \(c\text{,}\) denoted by \(\abs{c}\text{.}\) Because distance is never negative, the absolute value of a number is always positive (or zero). Thus, \(\abs{6}= 6\) and \(\abs{-6} = 6\text{.}\) In general, we define the absolute value of a number \(x\) as follows.
This definition is called piecewise, because the formula has two pieces. It says that the absolute value of a positive number (or zero) is the same as the number. To find the absolute value of a negative number, we take the opposite of the number, which is then positive. For instance,
Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
Consider the first pair of graphs. You have already studied the graph of \(f(x)=x^2\text{,}\) the basic parabola. Compare that graph with the graph of \(g(x)=x^3\text{.}\) Notice several differences in the shape of the two graphs. Once you have a good idea of the shape of a graph, up can make a quick sketch with just a few "guide points." For these two graphs, complete a short table of values to find useful guide points:
The next pair of graphs are \(f(x)=\sqrt{x}\) and \(g(x)=\sqrt[3]{x}\text{.}\) Once again, notice the differences in the two graphs. For example, we cannot take the square root of a negative number, but we can take its cube root. How is this reflected in the graphs?
The next pair of functions, \(f(x)=\dfrac{1}{x}\) and \(g(x)=\dfrac{1}{x^2}\text{,}\) are both undefined at \(x=0\text{,}\) so thier graphs do not include any points with \(x\)-coordinate zero. For very small positive values of \(x\text{,}\) both \(f(x)\) and \(g(x)\) get very large. As \(x\) gets closer to zero, the graphs approach the vertical line \(x=0\) (the \(y\)-axis). This line is called a vertical asymptote for the graph.
Also, notice that for very large values of \(x\text{,}\) both \(f(x)\) and \(g(x)\) get very close to zero. Their graphs approach the horizontal line \(y=0\) (the \(x\)-axis). This line is called the horizontal asymptote for the graph.
Finally, compare the familiar graph of \(f(x)=x\) with the graph of \(g(x)=\abs{x}\text{.}\) The piecewise definition of \(\abs{x}\) means that we graph \(y=x\) in the first quadrant (where \(x \ge 0\)), and \(y=-x\) in the first quadrant \(x \lt 0\)). The result is the V-shaped graph shown below.
Because they are fundamental to further study of mathematics and its applications, you should become familiar with the properties of these eight graphs, and be able to sketch them easily from memory, using their basic shapes and a few guidepoints.
For Problems 11β16, sketch the graph of the function by hand, paying attention to the shape of the graph. Carefully plot at least three βguide pointsβ to ensure accuracy. If possible, plot the points with \(x\)-coordinates \(-1, ~0,\) and \(1\text{.}\)
Use your calculator to graph \(f(x)=\sqrt{x}\) and \(g(x)=\sqrt[3]{x}\) on the same axes for \(0 \le x \le 1\text{.}\) Which function is greater on that interval?
Use your calculator to graph \(f(x)=\dfrac{1}{x}\) and \(g(x)=\dfrac{1}{x^2}\) on the same axes for \(0 \le x \le 1\text{.}\) Which function is greater on that interval?
Use your calculator to graph \(f(x)=\dfrac{1}{x}\) and \(g(x)=\dfrac{1}{x^2}\) on the same axes for \(1 \le x \le 4\text{.}\) Which function is greater on that interval?
For Problems 21β26, graph the functions in the same window on your calculator. Describe how the graphs in parts (b) and (c) are different from the basic graph.