# SAT 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

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

–1/2

1/2

2

3

0

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, 2, 3, 4}
{5, 6, 7, 8}
{2, 4, 6, 8}
{1, 3, 5, 7}
the inverse of the relation does not exist
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.

The inverse of the relation does not exist.

Explanation:

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).

does not exist

Explanation:

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:

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?

Explanation:

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 #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).

does not exist

does not exist

Explanation:

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 .