### All PSAT Math Resources

## 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* = *x*^{2} + 2?

**Possible Answers:**

{2}

all real numbers

{–2, 2}

*y* ≥ 2

undefined

**Correct answer:**

*y* ≥ 2

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* = *x*^{2} + 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* = *x*^{2} + 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* = (2*x –* 1) / (*x*^{2} – 6*x* + 9) ?

**Possible Answers:**

3

0

2

1/2

–1/2

**Correct answer:**

3

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 #1 : 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.

**Possible Answers:**

**Correct answer:**{1, 3, 5, 7}

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* = *x*^{2} + 2?

**Possible Answers:**

{2}

all real numbers

undefined

*y* ≥ 2

{–2, 2}

**Correct answer:**

*y* ≥ 2

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* = *x*^{2} + 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* = *x*^{2} + 2 will never produce a number less than 2. Therefore the range is *y* ≥ 2.

### Example Question #2 : 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 ?

**Possible Answers:**

**Correct answer:**

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 .

The answer is .

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

If , then find

**Possible Answers:**

**Correct answer:**

is the same as .

To find the inverse simply exchange and and solve for .

So we get which leads to .

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

If , then which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

### Example Question #2 : 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.

**Possible Answers:**

The inverse of the relation does not exist.

**Correct answer:**

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

For the original relation, the range is: .

Thus, the domain for the inverse relation will also be .