### All SAT Math Resources

## Example Questions

### 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:**

1/2

2

3

0

–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}

{–2, 2}

*y* ≥ 2

undefined

all real numbers

**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

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 #1 : 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 .

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

Define , restricting the domain of the function to .

Determine (you need not determine its domain restriction).

**Possible Answers:**

does not exist

**Correct answer:**

First, we must determine whether exists.

A quadratic function has a parabola as its graph; this graph decreases, then increases (or vice versa), with a vertex at which the change takes place.

exists if and only if, if , then - or, equivalently, if there does *not *exist and such that , but . This will happen on any interval on which the graph of constantly increases or constantly decreases, but if the graph changes direction on an interval, there will be such that on this interval. The key is therefore to determine whether the interval to which the domain is restricted contains the vertex.

The -coordinate of the vertex of the parabola of the function

is .

The -coordinate of the vertex of the parabola of can be found by setting :

.

The vertex of the graph of *without* its domain restriction is at the point with -coordinate 2. However, . Therefore, the domain at which is restricted does not include the vertex, and exists on this domain.

To determine the inverse of , first, rewrite in vertex form

, the same as in the standard form.

The graph of , if unrestricted, would have a vertex with -coordinate 2, and -coordinate

.

Therefore, .

The vertex form of is therefore

To find , first replace with :

Switch and :

Solve for . First, add 8 to both sides:

Take the square root of both sides:

Add 2 to both sides

Replace with :

Either or

The domain of is the set of nonpositive numbers; this is consequently the range of . can only have positive values, so the only possible choice for is .

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

Define a function .

It is desired that is domain be restricted so that have an inverse. Which of these domain restrictions would *not* achieve that goal?

**Possible Answers:**

**Correct answer:**

A quadratic function has a parabola as its graph; this graph decreases, then increases (or vice versa), with a vertex at which the change takes place.

exists if and only if, if , then - or, equivalently, if there does *not *exist and such that , but . This will happen on any interval on which the graph of constantly increases or constantly decreases, but if the graph changes direction on an interval, there will be such that on this interval. The key is therefore to identify the interval that contains the vertex.

The -coordinate of the vertex of the parabola of the function

is .

The -coordinate of the vertex of the parabola of can be found by setting :

.

Of the five intervals in the choices,

,

so cannot exist if is restricted to this interval. This is the correct choice.

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

Define , restricting the domain of the function to .

Determine (you need not determine its domain restriction).

**Possible Answers:**

does not exist

**Correct answer:**

does not exist

First, we must determine whether exists.

A quadratic function has a parabola as its graph; this graph changes direction (downward to upward, or vice versa) at a given point called the vertex.

exists on a given domain interval if and only if there does *not *exist and on this domain such that , but . This will happen if the graph changes direction on the domain interval. The key is therefore to determine whether the given domain interval includes the vertex.

The -coordinate of the vertex of the parabola of the function

is .

The -coordinate of the vertex of the parabola of can be found by setting :

.

The vertex of the graph of *without* its domain restriction is at the point with -coordinate 8. Since , the vertex is in the interior of the domain; as a consequence, the graph of changes direction on the interval, and does not exist on .