Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
354
Domain and Range of a Function
What can you plug in? What comes out?
Note: Problems 16.9–16.11 refer to the function .
16.9 Graph f(x) using a table of values.
Construct a table of values.
As illustrated by Figure 16-5, the graph of f(x) passes through the coordinate
pairs generated by the table of values: (–4,0), (–3,1), , , (0,2),
and .
Figure 16-5: The graph of .
This time
the book doesn’t
give you x-values
to plug into the
function. You don’t
need them—just
make sure you use
enough points to
visualize the
shape of the
graph.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
355
Note: Problems 16.9–16.11 refer to the function .
16.10 Identify the domain of f(x).
The domain of a function is the set of real numbers for which f(x) is defined.
A square root, and in fact any root with an even index, is defined only when the
radicand is nonnegative. Therefore, f(x) is defined only when x + 4 ≥ 0. Solve
the inequality.
Another method by which to identify the domain of a function is via its graph.
Consider the graph of f(x) in Figure 16-5. Any vertical line drawn on the
coordinate plane that intersects the graph represents a member of the domain.
The vertical line x = –4 is the leftmost vertical line that intersects f(x), so
x = –4 belongs to the domain of the function. All vertical lines to the right of
x = –4 intersect f(x) as well, so the domain of f(x) is x ≥ –4.
Note: Problems 16.9–16.11 refer to the function .
16.11 Identify the range of f(x).
Problem 16.10 states that vertical lines intersecting the graph of a function
represent members of that function’s domain. Similarly, horizontal lines that
intersect a graph represent members of the range. Consider the graph of f(x)
in Figure 16-5. The horizontal line y = 0 and all of the horizontal lines above it
intersect f(x), so the range of the function is f(x) ≥ 0.
In other
words, the
domain is the
collection of
numbers you’re
allowed to plug
into the
function.
The radicand
is the expression
inside the radical,
in this case x + 4.
A function’s
range is the
collection of all its
possible outputs. For
instance, if r(5) = –12
for some function r(x),
then 5 is a member of
the domain and –12
is a member of the
range.
It might
not look like
the horizontal
lines y = 4, y = 5,
and y = 6 intersect
the graph, but they do.
A square root graph
increases steadily (if
slowly) forever and
ever as the graph
travels right.
Instead of
writing y ≥ 0 for the range, use
f(x) ≥ 0 instead. Technically, the function
doesn’t contain y, so the
answer shouldn’t contain y either.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
356
Note: Problems 16.12–16.14 refer to the function .
16.12 Graph g(x) using a table of values.
Construct a table of values.
As illustrated by Figure 16-6, the graph of g(x) passes through the coordinate
pairs generated by the table of values: (0,5), (1,4), (2,3), (3,2), (4,3), (5,4),
and (6,5).
Figure 16-6: The graph of .
Note: Problems 16.12–16.14 refer to the function .
16.13 Identify the domain of g(x).
Consider the graph of g(x) in Figure 16-6. Any vertical line drawn on the
coordinate plane will intersect the graph. Therefore, all real numbers comprise
the domain of g(x).
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
357
Note: Problems 16.12–16.14 refer to the function .
16.14 Identify the range of g(x).
Consider the graph of g(x) in Figure 16-6. The horizontal line y = 2 intersects
the graph, as do all the horizontal lines above y = 2. Therefore, the range is
g(x) ≥ 2.
16.15 Identify the domain of .
Rational functions (that is, functions defined as fractions) are defined for all
real numbers, except values that make their denominators equal zero. Set the
denominator of j(x) equal to 0 and solve the equation.
Function j(x) is undefined when x = 8. Therefore, the domain of j(x) is all real
numbers except 8.
16.16 Identify the domain of .
A square root is undefined when its radicand is negative. Therefore, the
expressions x – 9 and 4x + 3 both must be greater than or equal to 0.
Function h(x) is only defined when x ≥ 9 and , so the domain of h(x) is
x ≥ 9.
16.17 Identify the domain of .
The domain of a rational function consists of all real numbers except the values
that cause the denominator to equal zero. Set the denominator equal to zero
and solve the equation to identify the values that must be explicitly excluded
from the domain.
x
2
– x – 4 = 0
Solve the equation using the quadratic formula.
At the
point (3,2)
Problems
16.10 and
16.13 use the
graph of a function
to nd its domain,
and Problems 16.15–
16.20 show you how to
identify the domain
without a graph. Range
is a different story—
graphing is often the
best way to identify
the range, especially
when the functions
are complicated.
If you
plug x = 8
into j(x), you get
.
You’re not allowed to
divide by 0, so x = 8
is excluded from
the domain.
If a number
is greater than
or equal to 9,
it’s automatically
greater than or
equal to
, so
there’s no need
to include the
fraction in the
answer.
Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
358
The domain of p(x) is all real numbers except and .
16.18 Identify the domain of .
Function k(x) is undefined when the radicand in the numerator is negative, so
identify the values of x for which 2 – x is nonnegative.
The function is also undefined when the denominator is equal to zero. Identify
those values to explicitly exclude them from the domain.
The denominator of k(x) equals zero when or . The domain of
the function is x ≤ 2, .
16.19 Identify the domain of .
Function q(x) is undefined when the denominator is equal to zero or when the
radicand is negative.
To identify values of x for which the radicand is positive, solve the inequality
3x
2
+ 8x – 3 > 0 using the technique described in Problems 14.35–14.43. Begin
by solving the equation 3x
2
+ 8x – 3 = 0 to identify the critical numbers of the
radicand.
When you
reverse the sides
of an inequality, you
have to reverse the
inequality sign as
well, changing ≥
to ≤.
,
and according to
the numerator, only
numbers less than or
equal to 2 belong in the
domain.
is less
than 2, so you have to
exclude it, but
is excluded
automatically.
When you’re
nding the
domain, the two big
things you look for are
zero in the denominator
and negatives inside
radicals (when the
radicals have an even
index—negatives inside
a cube root, for
example, are ne).
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