Geometry › How to graph a function
Give the domain of the function
The set of all real numbers
The square root of a real number is defined only for nonnegative radicands; therefore, the domain of is exactly those values for which the radicand
is nonnegative. Solve the inequality:
The domain of is
.
The chord of a central angle of a circle with area
has what length?
The radius of a circle with area
can be found as follows:
The circle, the central angle, and the chord are shown below:
By way of the Isosceles Triangle Theorem, can be proved equilateral, so
, the correct response.
Give the domain of the function
The set of all real numbers
The domain of any polynomial function, such as , is the set of real numbers, as a polynomial can be evaluated for any real value of
.
Which of the following are the equations of the vertical asymptotes of the graph of ?
(a)
(b)
(b) only
(a) only
Both (a) and (b)
Neither (a) nor (b)
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the denominator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as
by finding two integers with sum 6 and product 5. By trial and error, these integers can be found to be 1 and 5, so
Therefore, can be rewritten as
Set the denominator equal to 0 and solve for :
By the Zero Factor Principle,
or
Therefore, the binomial factor can be cancelled, and the function can be rewritten as
If , then
, so the denominator has only this one zero, and the only vertical asymptote is the line of the equation
.
True or false: The graph of has as a vertical asymptote the graph of the equation
.
False
True
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator. It is a quadratic trinomial with lead term , so look to factor
by using the grouping technique. We try finding two integers whose sum is and whose product is
; with some trial and error we find that these are
and
, so:
Break the linear term:
Regroup:
Factor the GCF twice:
Therefore, can be rewritten as
Cancel the common factor from both halves; the function can be rewritten as
Set the denominator equal to 0 and solve for :
The graph of therefore has one vertical asymptotes, the line of the equations
. The line of the equation
is not a vertical asymptote.
What is the domain of ?
all real numbers
The domain of the function specifies the values that can take. Here,
is defined for every value of
, so the domain is all real numbers.
Define
What is the natural domain of ?
The radical in and of itself does not restrict the domain, since every real number has a real cube root. However, since the expression is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which
27 is the only number excluded from the domain.
Give the domain of the function
The set of all real numbers
The function is defined for those values of
for which the radicand is nonnegative - that is, for which
Subtract 25 from both sides:
Since the square root of a real number is always nonnegative,
for all real numbers . Since the radicand is always positive, this makes the domain of
the set of all real numbers.
Give the -coordinate(s) of the
-intercept(s) of the graph of the function
The graph of has no
-intercept.
The -intercept(s) of the graph of
are the point(s) at which it intersects the
-axis. The
-coordinate of each is 0; their
-coordinate(s) are those value(s) of
for which
, so set up, and solve for
, the equation:
Add to both sides:
Multiply both sides by 2:
,
the correct choice.
Give the -coordinate of the
-intercept of the graph of the function
The graph of has no
-intercept.
The -intercept of the graph of
is the point at which it intersects the
-axis. Its
-coordinate is 0,; its
-coordinate is
, which can be found by substituting 0 for
in the definition:
However, does not have a real value. Therefore, the graph of
has no
-intercept.