Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
365
Because g(x) = , g(x) is an even function.
Fundamental Function Graphs
The graphs you need to understand most
16.26 Graph the function f(x) = x
2
using a table of values. Identify the domain
and range of the function, as well as the type of symmetry (if any) the graph
exhibits.
Figure 16-14 illustrates the graph of f(x) = x
2
and the table of values used to
generate the graph.
Figure 16-14: The graph of a quadratic function is a parabola.
The domain of f(x) is all real numbers; the range is f(x) ≥ 0, as squaring a real
number always produces a nonnegative result. The graph is symmetric about
the y-axis, as squaring a real number x and its opposite –x both produce the
same result: f(–x) = f(x).
Even func-
tions usually
have even powers
in them like this, and
odd functions usually
have odd powers.
However, the powers
alone are not enough
evidence to classify
the function as
even or odd.
You should
memorize the
graphs in Problems
16.26–16.30. If you
have to use a table of
values to draw these
ve graphs every
time, the rest of the
chapter will take a
whole lot longer.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
366
16.27 Graph the function g(x) = x
3
using a table of values. Identify the domain
and range of the function, as well as the type of symmetry (if any) the graph
exhibits.
Figure 16-15 illustrates the graph of f(x) = x
3
and the table of values used to
generate the graph.
Figure 16-15: The graph of the cubic function, g(x) = x
3
.
The domain and range of g(x) both consist of all real numbers—you may cube
both positive and negative real numbers, and doing so produces both positive
and negative results, respectively. The graph is origin-symmetric, as cubing a
number x and its opposite –x produce opposite results: g(–x) = –g(x).
16.28 Graph the function using a table of values. Identify the domain
and range of the function, as well as the type of symmetry (if any) the graph
exhibits.
Figure 16-16 illustrates the graph of and the table of values used to
generate the graph.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
367
Figure 16-16: The graph of the absolute value function has a cusp—a sharp point or
corner—at x = 0.
The domain of h(x) is all real numbers, and the range is h(x) ≥ 0—every real
number has an absolute value, and it is a nonnegative number. The graph of
h(x) is symmetric about the y-axis, as the absolute value of a real number x and
its opposite –x are equal: h(–x) = h(x).
16.29 Graph the function using a table of values. Identify the domain
and range of the function, as well as the type of symmetry (if any) the graph
exhibits.
Figure 16-17 illustrates the graph of and the table of values used to
generate the graph.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
368
Figure 16-17: The graph of the square root function is located within the first quadrant
of the coordinate plane.
The domain of j(x) is x ≥ 0, and the range is j(x) ≥ 0—you can only take the
square root of a nonnegative number, and the result is a nonnegative number.
The graph exhibits no x-, y-, or origin-symmetry.
16.30 Graph the function using a table of values. Identify the domain
and range of the function, as well as the type of symmetry (if any) the graph
exhibits.
Figure 16-18 illustrates the graph of and the table of values used to
generate the graph.
Figure 16-18: The graph of k(x) intersects neither the x-axis nor the y-axis.
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