All Advanced Geometry Resources
Example Questions
Example Question #2 : How To Graph A Function
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, along with
, which bisects isoscelesWe concentrate on
, a 30-60-90 triangle. By the 30-60-90 Theorem,
and
The chord
has length twice this, or
Example Question #211 : Advanced Geometry
The chord of a
central angle of a circle with circumference has what length?
A circle with circumference
has as its radius.
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.Example Question #1 : How To Graph A Function
What is the domain of
?
all real numbers
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.Example Question #212 : Advanced Geometry
What is the domain of
?
To find the domain, we need to decide which values
can take. The is under a square root sign, so cannot be negative. can, however, be 0, because we can take the square root of zero. Therefore the domain is .Example Question #3 : Graphing A Function
What is the domain of the function
?
To find the domain, we must find the interval on which
is defined. We know that the expression under the radical must be positive or 0, so is defined when . This occurs when and . In interval notation, the domain is .Example Question #4 : Graphing A Function
Define the functions
and as follows:
What is the domain of the function
?
The domain of
is the intersection of the domains of and . and are each restricted to all values of that allow the radicand to be nonnegative - that is,, or
Since the domains of
and are the same, the domain of is also the same. In interval form the domain of isExample Question #4 : How To Graph A Function
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.
Example Question #5 : Graphing A Function
Define
What is the natural domain of
?
Since the expression
is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which . We solve for by factoring the polynomial, which we can do as follows:
Replacing the question marks with integers whose product is
and whose sum is 3:
Therefore, the domain excludes these two values of
.Example Question #1 : How To Graph A Function
Define
.What is the natural domain of
?
The only restriction on the domain of
is that the denominator cannot be 0. We set the denominator to 0 and solve for to find the excluded values:
The domain is the set of all real numbers except those two - that is,
.
Example Question #121 : Coordinate Geometry
The figure above shows the graph of y = f(x). Which of the following is the graph of y = |f(x)|?
One of the properties of taking an absolute value of a function is that the values are all made positive. The values themselves do not change; only their signs do. In this graph, none of the y-values are negative, so none of them would change. Thus the two graphs should be identical.