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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