# PSAT 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

What is the range of the function y = x2 + 2?

y ≥ 2

{–2, 2}

all real numbers

undefined

{2}

y ≥ 2

Explanation:

The range of a function is the set of y-values that a function can take. First let's find the domain. The domain is the set of x-values that the function can take. Here the domain is all real numbers because no x-value will make this function undefined. (Dividing by 0 is an example of an operation that would make the function undefined.)

So if any value of x can be plugged into y = x2 + 2, can y take any value also? Not quite! The smallest value that y can ever be is 2. No matter what value of x is plugged in, y = x2 + 2 will never produce a number less than 2. Therefore the range is y ≥ 2.

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

Which of the following values of x is not in the domain of the function y = (2x – 1) / (x2 – 6x + 9) ?

1/2

0

3

–1/2

2

3

Explanation:

Values of x that make the denominator equal zero are not included in the domain. The denominator can be simplified to (x – 3)2, so the value that makes it zero is 3.

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

Given the relation below:

{(1, 2), (3, 4), (5, 6), (7, 8)}

Find the range of the inverse of the relation.

{1, 3, 5, 7}
{1, 2, 3, 4}
{2, 4, 6, 8}
the inverse of the relation does not exist
{5, 6, 7, 8}
Correct answer: {1, 3, 5, 7}
Explanation:

The domain of a relation is the same as the range of the inverse of the relation.  In other words, the x-values of the relation are the y-values of the inverse.

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

What is the range of the function y = x2 + 2?

{2}

undefined

y ≥ 2

all real numbers

{–2, 2}

y ≥ 2

Explanation:

The range of a function is the set of y-values that a function can take. First let's find the domain. The domain is the set of x-values that the function can take. Here the domain is all real numbers because no x-value will make this function undefined. (Dividing by 0 is an example of an operation that would make the function undefined.)

So if any value of x can be plugged into y = x2 + 2, can y take any value also? Not quite! The smallest value that y can ever be is 2. No matter what value of x is plugged in, y = x2 + 2 will never produce a number less than 2. Therefore the range is y ≥ 2.

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

What is the smallest value that belongs to the range of the function ?

Explanation:

We need to be careful here not to confuse the domain and range of a function. The problem specifically concerns the range of the function, which is the set of possible numbers of . It can be helpful to think of the range as all the possible y-values we could have on the points on the graph of .

Notice that  has  in its equation. Whenever we have an absolute value of some quantity, the result will always be equal to or greater than zero. In other words, |4-x| 0. We are asked to find the smallest value in the range of , so let's consider the smallest value of , which would have to be zero. Let's see what would happen to  if .

This means that when , . Let's see what happens when  gets larger. For example, let's let .

As we can see, as  gets larger, so does . We want  to be as small as possible, so we are going to want  to be equal to zero. And, as we already determiend,  equals  when .

### Example Question #5 : How To Find Domain And Range Of The Inverse Of A Relation

If , then find

Explanation:

is the same as

To find the inverse simply exchange  and  and solve for

So we get  which leads to .

### Example Question #6 : How To Find Domain And Range Of The Inverse Of A Relation

If , then which of the following is equal to ?

Explanation:

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

Given the relation below, identify the domain of the inverse of the relation.