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

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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 functiony=x²+ 2?

Possible Answers:

{–2, 2}

all real numbers

undefined

{2}

y≥ 2

Correct answer:

y≥ 2

Explanation:

The range of a function is the set ofy-values that a function can take. First let's find the domain. The domain is the set ofx-values that the function can take. Here the domain is all real numbers because nox-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 ofxcan be plugged intoy=x²+ 2, canytake any value also? Not quite! The smallest value thatycan ever be is 2. No matter what value ofxis plugged in,y=x²+ 2 will never produce a number less than 2. Therefore the range isy≥ 2.

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

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

Possible Answers:

1/2

–1/2

Correct answer:

Explanation:

Values ofxthat 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.

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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.

Possible Answers:

the inverse of the relation does not exist

{5, 6, 7, 8}

{2, 4, 6, 8}

{1, 2, 3, 4}

{1, 3, 5, 7}

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.

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Example Question #3 :How To Find Domain And Range Of The Inverse Of A Relation

What is the range of the functiony=x²+ 2?

Possible Answers:

undefined

all real numbers

{–2, 2}

y≥ 2

{2}

Correct answer:

y≥ 2

Explanation:

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

Possible Answers:

$\dpi{100} -2$

$\dpi{100} -4$

$\dpi{100} 4$

$\dpi{100} 0$

$\dpi{100} 2$

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 $\dpi{100} -2$ .

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

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

Possible Answers:

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

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

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

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

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

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 $y=x+3$ .

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

If $f(x)=\frac{x}{x-3}$ , then which of the following is equal to $f^{-1}(x)$ ?

Possible Answers:

$\frac{-x}{x-3}$

$\frac{-3x}{x+1}$

$\frac{-x}{x+3}$

$\frac{3x}{x-1}$

Correct answer:

$\frac{3x}{x-1}$

Explanation:

Inversef2

Inverse3

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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.

$\left \{(1, 2), (3, 4), (5, 6), (7, 8){ \right \}$

Possible Answers:

$\left \{ 1,\ 3,\ 5,\ 7 \right \}$

$\left \{ 2,\ 4,\ 6,\ 8 \right \}$

The inverse of the relation does not exist.

$\left \{ 5,\ 6,\ 7,\ 8 \right \}$

$\left \{ 1,\ 2,\ 3,\ 4 \right \}$

Correct answer:

$\left \{ 2,\ 4,\ 6,\ 8 \right \}$

Explanation:

$\left \{(1, 2), (3, 4), (5, 6), (7, 8){ \right \}$

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: $\left \{ 2,\ 4,\ 6,\ 8 \right \}$ .

Thus, the domain for the inverse relation will also be $\left \{ 2,\ 4,\ 6,\ 8 \right \}$ .

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

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

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

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

Example Question #3 :How To Find Domain And Range Of The Inverse Of A Relation

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

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

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

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

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