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

Study concepts, example questions & explanations for PSAT Math

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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?

Possible Answers:

all real numbers

{–2, 2}

undefined

{2}

y ≥ 2

Correct answer:

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) ?

Possible Answers:

1/2

–1/2

0

3

2

Correct answer:

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 #354 : Algebra

Given the relation below:

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

Find the range of the inverse of the relation.

Possible Answers:
{1, 3, 5, 7}
{2, 4, 6, 8}
{1, 2, 3, 4}
{5, 6, 7, 8}
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?

Possible Answers:

undefined

y ≥ 2

all real numbers

{–2, 2}

{2}

Correct answer:

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 #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 f(x)=2|4-x|-2 ?

Possible Answers:

\dpi{100} 2

\dpi{100} -4

\dpi{100} 0

\dpi{100} 4

\dpi{100} -2

Correct answer:

\dpi{100} -2

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 \dpi{100} f(x). 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 \dpi{100} f(x).

Notice that \dpi{100} f(x) has \dpi{100} |4-x| 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| \geq 0. We are asked to find the smallest value in the range of \dpi{100} f(x), so let's consider the smallest value of \dpi{100} |4-x|, which would have to be zero. Let's see what would happen to \dpi{100} f(x) if \dpi{100} |4-x|=0.

\dpi{100} f(x)=2(0)-2=0-2=-2

This means that when \dpi{100} |4-x|=0, \dpi{100} f(x)=-2. Let's see what happens when \dpi{100} |4-x| gets larger. For example, let's let \dpi{100} |4-x|=3.

\dpi{100} f(x)=2(3)-2=4

As we can see, as \dpi{100} |4-x| gets larger, so does \dpi{100} f(x). We want \dpi{100} f(x) to be as small as possible, so we are going to want \dpi{100} |4-x| to be equal to zero. And, as we already determiend, \dpi{100} f(x) equals \dpi{100} -2 when \dpi{100} |4-x|=0.

The answer is \dpi{100} -2.

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

If f(x) = x - 3, then find f^{-1}(x)

Possible Answers:

f^{-1}(x)=\frac{1}{x-3}

f^{-1}(x)=x-3

f^{-1}(x)=x+3

f^{-1}(x)=3-x

f^{-1}(x)=3x

Correct answer:

f^{-1}(x)=x+3

Explanation:

f(x) = x - 3 is the same as y= x - 3

To find the inverse simply exchange x and y and solve for y

So we get x=y-3 which leads to y=x+3.

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:

Explanation:

Inversef2

Inverse3

Example Question #5 : 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:

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 .

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