All Advanced Geometry Resources
Example Questions
Example Question #31 : How To Graph A Function
Give the domain of the function
The set of all real numbers
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 .
Example Question #32 : How To Graph A Function
Give the domain of the function
The set of all real numbers
The set of all real numbers
There is no restriction on the value of , as a cube root can be taken of any real number, regardless of sign. Since is not restricted in value, neither is , and the domain of is the set of real numbers.
Example Question #153 : Graphing
Give the domain of the function
.
The set of all real numbers
The set of all real numbers
As a rational function, has as its domain the set of all values for which the denominator is not equal to 0. Solve the equation
However, there is no real number for which this equation holds, as the square of any such number must be positive. Therefore, the domain of does not exclude any real values, and the domain is the set of all real numbers.
Example Question #154 : Graphing
Give the -coordinate of the -intercept of the graph of the function
The graph of has no -intercept.
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.
Example Question #155 : Graphing
Give the -coordinate(s) of the -intercept(s) of the graph of the function
.
and
The graph of has no -intercept.
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:
For this quantity to be equal to 0, it must hold that the numerator is equal to 0, so
This, however, is identically false, so cannot be equal to 0 for any value of . It follows that the graph of has no -intercept.
Example Question #156 : Graphing
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:
For this quantity to be equal to 0, it must hold that the numerator is equal to 0, so
in which case
,
the correct choice.
Example Question #157 : Graphing
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:
For this quantity to be equal to 0, it must hold that the numerator is equal to 0, so
and
,
the correct choice.
Example Question #33 : How To Graph A Function
Give the domain of the function
The set of all real numbers
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.
Example Question #155 : Graphing
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)
(b) only
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 .
Example Question #241 : Advanced Geometry
True or false: The graph of has as a horizontal asymptote the graph of the equation .
True
False
True
is a rational function whose denominator polynomial has degree greater than that of its numerator polynomial (2 and 1, respectively). The graph of such a function has as its horizontal asymptote the line of the equation .