Advanced Geometry : Advanced Geometry

Study concepts, example questions & explanations for Advanced Geometry

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Example Questions

Example Question #31 : How To Graph A Function

Give the domain of the function

Possible Answers:

The set of all real numbers

Correct answer:

The set of all real numbers

Explanation:

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

Possible Answers:

The set of all real numbers

Correct answer:

The set of all real numbers

Explanation:

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

.

Possible Answers:

The set of all real numbers

Correct answer:

The set of all real numbers

Explanation:

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

Possible Answers:

The graph of  has no -intercept.

Correct answer:

The graph of  has no -intercept.

Explanation:

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

.

Possible Answers:

 and 

The graph of  has no -intercept.

Correct answer:

The graph of  has no -intercept.

Explanation:

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

.

Possible Answers:

The graph of  has no -intercept.

Correct answer:

Explanation:

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

.

Possible Answers:

The graph of  has no -intercept.

Correct answer:

Explanation:

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

Possible Answers:

The set of all real numbers

Correct answer:

The set of all real numbers

Explanation:

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) 

Possible Answers:

(b) only

(a) only

Both (a) and (b) 

Neither (a) nor (b) 

Correct answer:

(b) only

Explanation:

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 .

Possible Answers:

True

False

Correct answer:

True

Explanation:

 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 .

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