How to find domain and range of the inverse of a relation - ACT Math
Card 0 of 54
Define function 
as follows:
On which of the following restrictions of the domain of 
would 
not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
is a quadratic function, so its graph is a parabola. The key is to find the 
-intercept of the vertex of the parabola, which can be found by completing the square:
The vertex happens at 
, so the interval which contains this value will have at least one pair 
such that 
. The correct choice is 
.
The vertex happens at
Compare your answer with the correct one above
Define function 
as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
has a sinusoidal wave as its graph, with period 
; it begins at a relative maximum of 
and has a relative maximum or minimum every 
units. Therefore, any interval containing an integer multiple of 
will have at least two distinct values 
such that 
.
The only interval among the choices that includes a multiple of 
is 
:
.
This is the correct choice.
The only interval among the choices that includes a multiple of
This is the correct choice.
Compare your answer with the correct one above
Define function 
as follows:
In which of the following ways could the domain of 
be restricted so that 
does not have an inverse?
Define function
In which of the following ways could the domain of
If 
, then 
. By the addition property of inequality, if 
, then 
. Therefore, if 
, 
.
Consequently, there can be no 
such that 
, regardless of how the domain is restricted. 
will have an inverse regardless of any domain restriction.
If
Consequently, there can be no
Compare your answer with the correct one above
Define function 
as follows:
Suppose the domain of 
were to be restricted so that 
could have an inverse. Which of the following restrictions would not give 
an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are 
.
has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair such that . Since a cubic polynomial has two "arms", one going up and one going down, 
will increase as 
increases in the other four intervals. 
is the correct choice.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are
Compare your answer with the correct one above
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
Compare your answer with the correct one above
Define function 
as follows:
Suppose the domain of 
were to be restricted so that 
could have an inverse. Which of the following restrictions would not give 
an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period and phase shift 
units to the left. Its positive "peaks" and "valleys" begin at 
and occur every 
units.
Since 
includes one of these "peaks" or "valleys", it contains at least two distinct values such that . It is the correct choice.
Since
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
is a quadratic function, so its graph is a parabola. The key is to find the -intercept of the vertex of the parabola, which can be found by completing the square:
The vertex happens at , so the interval which contains this value will have at least one pair such that . The correct choice is .
The vertex happens at
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period ; it begins at a relative maximum of and has a relative maximum or minimum every units. Therefore, any interval containing an integer multiple of will have at least two distinct values such that .
The only interval among the choices that includes a multiple of is :
.
This is the correct choice.
The only interval among the choices that includes a multiple of
This is the correct choice.
Compare your answer with the correct one above
Define function as follows:
In which of the following ways could the domain of be restricted so that does not have an inverse?
Define function
In which of the following ways could the domain of
If , then . By the addition property of inequality, if , then . Therefore, if , .
Consequently, there can be no such that , regardless of how the domain is restricted. will have an inverse regardless of any domain restriction.
If
Consequently, there can be no
Compare your answer with the correct one above
Define function as follows:
Suppose the domain of were to be restricted so that could have an inverse. Which of the following restrictions would not give an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are .
has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair such that . Since a cubic polynomial has two "arms", one going up and one going down, will increase as increases in the other four intervals. is the correct choice.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are
Compare your answer with the correct one above
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
Compare your answer with the correct one above
Define function as follows:
Suppose the domain of were to be restricted so that could have an inverse. Which of the following restrictions would not give an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period and phase shift units to the left. Its positive "peaks" and "valleys" begin at and occur every units.
Since includes one of these "peaks" or "valleys", it contains at least two distinct values such that . It is the correct choice.
Since
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
is a quadratic function, so its graph is a parabola. The key is to find the -intercept of the vertex of the parabola, which can be found by completing the square:
The vertex happens at , so the interval which contains this value will have at least one pair such that . The correct choice is .
The vertex happens at
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period ; it begins at a relative maximum of and has a relative maximum or minimum every units. Therefore, any interval containing an integer multiple of will have at least two distinct values such that .
The only interval among the choices that includes a multiple of is :
.
This is the correct choice.
The only interval among the choices that includes a multiple of
This is the correct choice.
Compare your answer with the correct one above
Define function as follows:
In which of the following ways could the domain of be restricted so that does not have an inverse?
Define function
In which of the following ways could the domain of
If , then . By the addition property of inequality, if , then . Therefore, if , .
Consequently, there can be no such that , regardless of how the domain is restricted. will have an inverse regardless of any domain restriction.
If
Consequently, there can be no
Compare your answer with the correct one above
Define function as follows:
Suppose the domain of were to be restricted so that could have an inverse. Which of the following restrictions would not give an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are .
has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair such that . Since a cubic polynomial has two "arms", one going up and one going down, will increase as increases in the other four intervals. is the correct choice.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are
Compare your answer with the correct one above
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
Compare your answer with the correct one above
Define function as follows:
Suppose the domain of were to be restricted so that could have an inverse. Which of the following restrictions would not give an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period and phase shift units to the left. Its positive "peaks" and "valleys" begin at and occur every units.
Since includes one of these "peaks" or "valleys", it contains at least two distinct values such that . It is the correct choice.
Since
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
is a quadratic function, so its graph is a parabola. The key is to find the -intercept of the vertex of the parabola, which can be found by completing the square:
The vertex happens at , so the interval which contains this value will have at least one pair such that . The correct choice is .
The vertex happens at
Compare your answer with the correct one above
Define function as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period ; it begins at a relative maximum of and has a relative maximum or minimum every units. Therefore, any interval containing an integer multiple of will have at least two distinct values such that .
The only interval among the choices that includes a multiple of is :
.
This is the correct choice.
The only interval among the choices that includes a multiple of
This is the correct choice.
Compare your answer with the correct one above