Negative reciprocal may sound complicated, but once we’ve interpreted its concept, you will do it see how easy that is come apply and also find a number’s negative reciprocal. Why don’t us dissect the two words?

Negative and also reciprocal – this means that a number’s an unfavorable reciprocal is the result of multiply the number’s mutual by $mathbf-1$.

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As straightforward together its definition, an adverse reciprocals have actually a wide selection of applications that incorporate finding perpendicular slopes and also modeling real-world applications that exploit inverse relationships.

What is a an unfavorable reciprocal?

When handle with an unfavorable reciprocals, we’ll first recall what these 2 words stand for in math: an unfavorable and reciprocal.

$oldsymboldfracab ightarrow – dfracba$

We’ll slowly malfunction this form, and also by the finish of this article, you’ll definitely be able to understand what this represents.

Reciprocal

The reciprocal of a number or a role is the value or expression that results from reversing the numerator and also denominator’s places.

$oldsymboldfracab ightarrow dfracba$

Reciprocals are considered multiplicative inverses because they will always be 1 when we main point a number by its reciprocal.

$dfracab cdot dfracba=1$

Master your knowledge on reciprocals here.

Negative the a Number (or a Function)

The an unfavorable of a number or a function is the result of a number gift multiplied through -1. Stop say we have a fraction, $dfracba$, its an adverse counterpart will be $-dfracba$.

$oldsymbol-1 cdot dfracba = -dfracba$

Learn an ext about an unfavorable numbers here.

When we integrate these two concepts, we’ll have actually the negative reciprocal the a number. This way that an adverse reciprocals result from us taking the mutual of a number then uncover the an unfavorable value the the result.

Hence, we have $oldsymboldfracab ightarrow – dfracba$.

How come find an unfavorable reciprocal?

Now the we understand what negative reciprocals represent, exactly how do we manipulate various forms of expression to have actually their negative reciprocals?

Always start by switching the areas of the numerator and the denominator of the fraction.Once we have the reciprocal, multiply the an outcome by $mathbf-1$.

We’ve created quick travel guide you can note as soon as working with different varieties of numbers and also expressions.

Let’s start by finding out to find the negative reciprocal of a fraction, $dfracab$, whereby $b eq 0$.

Take keep in mind that because that the negative reciprocals to exist, both $m$ and also $f(x)$ need to not be equal to $0$.

Excited to shot out difficulties involving an adverse reciprocals? First, let’s go ahead and also summarize what we’ve learned so far about an unfavorable reciprocals.

Summary the reciprocal meaning and properties

This expression to represent what wake up in detect the an adverse reciprocals: $oldsymboldfracab ightarrow – dfracba$.When offered a whole number or a function that is not rational, start by express the given as a fraction of 1.It is only possible for a constant or a function to have actually a negative reciprocal when both that is numerator and denominator are not same to $0$.A steep of a perpendicular line makes use of negative reciprocals.

That’s it. Make certain to store these needle in mind as soon as solving the troubles below.

Example 1

Complete the table listed below by detect the respective an adverse reciprocals of the following.

 Original Value Negative Reciprocal $dfrac12$ $-dfrac23$ $9$ $- 4dfrac17$

Solution

When finding the an adverse reciprocal, we start by convert the fraction’s numerator and also denominator places. Let’s work on the very first two items first: $dfrac12$ and $-dfrac23$.

Hence, their reciprocals are $dfrac21$ and $-dfrac32$.

For each value, main point $-1$ to discover the corresponding an adverse reciprocal.

-1 cdot dfrac21 = -2-1 cdot -dfrac32=dfrac32. We’ll actually use the same process for the last two rows, yet let’s first make certain we rewrite them in fraction form. The entirety number 9 can be composed as dfrac91 and the mixed number - 4dfrac17 can be written as -dfrac297. Once we have actually them in fraction forms, we have the right to now switch the locations of their corresponding numerators and denominators then multiplying the respective result by -1. eginaligneddfrac91 ightarrow dfrac19 ightarrow -dfrac19 endalignedeginaligned -dfrac297 ightarrow dfrac-729 ightarrow dfrac729 endaligned

Hence, we have actually the completed table as shown below.

 Original Value Negative Reciprocal $dfrac12$ $-2$ $-dfrac23$ $dfrac32$ $9$ $-dfrac19$ $- 4dfrac17$ $-dfrac729$

Example 2

Let $h(x)$ it is in the negative reciprocal the $f(x)$ for each of the adhering to functions. Uncover the $h(x)$. What space the restrictions for $x$ in every case?

a. $f(x) = dfrac1x – 1$

b. $f(x) = dfrac23(x +2)$

c. $f(x) = x^2 – 3x – 54$

Solution

We use the same procedure when recognize the an adverse reciprocals the functions.

a. This method that we begin by switching the locations of $1$ and also $x – 1$ to find the reciprocal of $f(x)$. We then multiply the an outcome by $-1$.

eginalignedh(x)&=-1cdot dfracx-11\&=-1cdot x – 1\&mathbf-x + 1 endaligned

Since $h(x)$ is a linear expression, it has actually not restrictions. The duty $f(x)$, however, should not have actually $x – 1 = 0$, therefore $mathbfx eq 0$.

b. We use the same procedure from a. Hence, we have actually $h(x)$ as presented below.

eginalignedh(x)&=-1cdot dfrac3(x+2)2\&=-1cdot dfrac3x+62\&=mathbf-dfrac3x+62 endaligned

The role $h(x)$ has actually a consistent as the denominator, so it has no restrictions for $x$. The function $f(x)$, however, can’t have actually $3(x + 2) = 0$, for this reason $mathbfx eq -2$.

c. To express $f(x)$ as a fraction by having $1$ together its denominator, therefore $f(x) = dfrac x^2 – 3x – 541$. Now, use the same process to discover the an unfavorable reciprocal, $h(x)$.

eginalignedh(x)&=-1cdot dfrac1x^2-3x-54\&=mathbf-dfrac1x^2-3x-54\ endaligned

Since $f(x)$ is a polynomial, it has no constraints for $x$. Its negative reciprocal, however, can’t have actually zero in that denominator. We can discover the restrictions for $h(x)$ by recognize the worths where $x^2 – 3x – 54$ is zero.

eginaligned x^2 -3x – 54&=0\(x – 9)(x + 6)&=0\x&=9\x&-6endaligned

This way that for $h(x)$ to be valid, $mathbfx eq -6,9$.

Example 3

The graph the the linear function, $f(x)$, is perpendicular come the graph of $h(x)$, i m sorry is additionally linear function. If $f(x)$ has actually a slope of $-dfrac23$, what is the slope of $h(x)$?

Solution

As we have actually mentioned in the discussion, finding an unfavorable reciprocals is an important when finding the slopes the perpendicular lines.

Since we have actually the slope of $f(x)$, we can uncover the slope of $h(x)$ by finding the negative reciprocal the $-dfrac23$.

eginalignedm_perp &= -1 cdot-dfrac32\&=dfrac32 endaligned

This means that the slope of $h(x)$ is $dfrac32$ for it to be perpendicular to $f(x)$.

Example 4

The negative reciprocal that $f(x)$ is $dfracx^2 – 2x – 5$. What is the expression because that $f(x)$?

Solution

This time, we’re provided the an adverse reciprocal. We need to discover the expression for $f(x)$ by reversing the steps:

We begin by multiply $-1$ earlier on the an unfavorable reciprocal to reverse the changes in the sign.Switch the locations of the an adverse reciprocal’s numerator and also denominator.

eginalignedf(x)&=-1cdot dfracx-5x^2-2\&=dfrac-x+5x^2-2 endaligned

This way that $mathbff(x) =dfrac-x+5x^2-2$.

Notice something around the steps? They space actually the same procedure because the an adverse reciprocal of a function’s an unfavorable reciprocal will be $mathbff(x)$.

Example 6

If a provided number is twenty-seven times bigger than its an adverse reciprocal’s square, find the number.

Solution

Let $n$ it is in the number we’re spring for, for this reason its an adverse can be expressed together $-dfrac1n$. Set up the equation representing the situation.

$n=27cdotleft(-dfrac1n ight )^2$

Simplify this equation by multiply both political parties of the equation through $n^2$ and also taking the cube root of both political parties of the equation.

eginalignedn&=dfrac27n^2\n^3&=27\sqrt<3>n^3 &=sqrt<3>27\n&=3 endaligned

This way that because that the number to fulfill the condition, it should be equal to 3.

Practice Questions

1. Complete the table listed below by finding the respective negative reciprocals of the following.

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 Original Value Negative Reciprocal $dfrac15$ $-dfrac611$ $-12$ $2dfrac38$

2. Allow $h(x)$ it is in the negative reciprocal of $f(x)$ for each that the adhering to functions. Discover the $h(x)$. What space the constraints for $x$ in every case?

a. $f(x) = dfrac23x – 5$b. $f(x) = dfracx2(x – 3)$c. $f(x) = x^2 – 7x – 30$d. $f(x) = 1 + dfrac1x – 2$

3. True or False? The reciprocal of the an adverse reciprocal the a role is same to the function itself.4. The graph the the direct function, $f(x)$, is perpendicular come the graph that $h(x)$, which is also linear function. If $f(x)$ has actually a steep of $-2dfrac15$, what is the slope of $h(x)$?5. The negative reciprocal that $f(x)$ is $dfracx^2 – 2x – 5$. What is the expression because that $f(x)$?6. Let $h(x)$ be the an unfavorable reciprocal the $f(x)$.a. What is the expression that $h(x)$ given that $f(x) = dfrac4x – 32$?b. What room the constraints for $x$ so the both $f(x)$ and $h(x)$ exist?c. Use your knowledge in graphing reciprocal functions to graph $h(x)$. Encompass the vertical and also horizontal asymptotes.7. If a offered number is sixty-four times larger than its negative reciprocal’s square, uncover the number.