# ACT Math : How to find domain and range of the inverse of a relation

## Example Questions

### Example Question #1 : How To Find Domain And Range Of The Inverse Of A Relation

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?

None of the other responses gives a correct answer.

Explanation:

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.

### Example Question #2 : How To Find Domain And Range Of The Inverse Of A Relation

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?

Explanation:

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.

### Example Question #1 : How To Find Domain And Range Of The Inverse Of A Relation

Define function  as follows:

On which of the following restrictions of the domain of  would  not exist?

None of the other responses gives a correct answer.

Explanation:

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 .

### Example Question #4 : How To Find Domain And Range Of The Inverse Of A Relation

Define function  as follows:

On which of the following restrictions of the domain of  would  not exist?

Explanation:

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.

### Example Question #2 : How To Find Domain And Range Of The Inverse Of A Relation

Define function  as follows:

In which of the following ways could the domain of  be restricted so that  does not have an inverse?

None of the other responses give a correct answer.

None of the other responses give a correct answer.

Explanation:

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.

### Example Question #1 : How To Find Domain And Range Of The Inverse Of A Relation

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?

If a fish is not a shark, then it is a carnivore.

If a fish is a shark, then it is a carnivore.

All fish are sharks.

If a fish is not a carnivore, then it is not a shark.

If a fish is not a shark, then it is not a carnivore.

If a fish is not a shark, then it is not a carnivore.

Explanation:

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.