Why are rational functions called "rational"?

Rational numbers are numbers that can be expressed as a fraction (a ratio) of two integers. Rational functions are also fractions (ratios) - with polynomials in the numerator and denominator.

How do you find the inverse of y=x+3/3 ?

y=(x+3)/3 3y=x+3 3y-3=x ∴x=3y-3

x-1 = -1/y+1, I got stuck on this part, I don't know how to get rid of a fraction with a denominator like that. help me please!

I assume y+1 is the denominator, not just y? We take your equation and multiply both sides by y+1. On the right, we get -1(y+1)/(y+1), and the (y+1)'s cancel. So we're left with (x-1)(y+1)= -1 Now we divide by (x+1), then subtract 1 to isolate y.

how do you find the inverse of F(x)= x/(x-1)

𝑦 = 𝑥/(𝑥 – 1) Swap the domain and range: 𝑥 = 𝑦/(𝑦 – 1) Now solve for the inverse: 𝑥𝑦 – 𝑥 = 𝑦 𝑥𝑦 – 𝑦 = 𝑥 𝑦(𝑥 – 1) = 𝑥 𝑦 = 𝑥/(𝑥 – 1) So interestingly, this function is its own inverse. Functions like this are called _involutions_. Comment if you have questions!

Main content

Course: Algebra (all content) > Unit 7

Lesson 20: Finding inverse functions (Algebra 2 level)

Finding inverse functions

Google Classroom

Learn how to find the formula of the inverse function of a given function. For example, find the inverse of f(x)=3x+2.

Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if

f

takes

a

b

, then the inverse,

f^{- 1}

, must take

b

a

Or in other words,

f (a) = b ⟺ f^{- 1} (b) = a

In this article we will learn how to find the formula of the inverse function when we have the formula of the original function.

Before we start...

In this lesson, we will find the inverse function of

f (x) = 3 x + 2

Before we do that, let's first think about how we would find

f^{- 1} (8)

To find

f^{- 1} (8)

, we need to find the input of

f

that corresponds to an output of

8

. This is because if

f^{- 1} (8) = x

, then by definition of inverses,

f (x) = 8

\begin{aligned} f (x) & = 3 x + 2 \\ 8 & = 3 x + 2 & Let f(x)=8 \\ 6 & = 3 x & Subtract 2 from both sides \\ 2 & = x & Divide both sides by 3 \end{aligned}

f (2) = 8

which means that

f^{- 1} (8) = 2

Finding inverse functions

We can generalize what we did above to find

f^{- 1} (y)

for any

y

To find

f^{- 1} (y)

, we can find the input of

f

that corresponds to an output of

y

. This is because if

f^{- 1} (y) = x

then by definition of inverses,

f (x) = y

\begin{aligned} f (x) & = 3 x + 2 \\ y & = 3 x + 2 & Let f(x)=y \\ y - 2 & = 3 x & Subtract 2 from both sides \\ \frac{y - 2}{3} & = x & Divide both sides by 3 \end{aligned}

f^{- 1} (y) = \frac{y - 2}{3}

Since the choice of the variable is arbitrary, we can write this as

f^{- 1} (x) = \frac{x - 2}{3}

Because inverse functions reverse the inputs and outputs, another way to find

f^{- 1}

is by switching

x

and

y

initially, then solving for

y

to write the inverse in function form.

\begin{aligned} f (x) & = 3 x + 2 \\ y & = 3 x + 2 & Replace f(x) with y \\ x & = 3 y + 2 & Switch x and y \\ x - 2 & = 3 y & Subtract 2 from both sides \\ \frac{x - 2}{3} & = y & Divide both sides by 3 \end{aligned}

f^{- 1} (x) = \frac{x - 2}{3}

Notice the similarities among the two methods. In each case, the equation is solved for the variable that is not isolated. The main difference involves when the variables are reversed. (The first method does this at the end while the second method does this initially.)

Check your understanding

1) Linear function

Find the inverse of $g (x) = 2 x - 5$ ‍.

g^{- 1} (x) =

2) Cubic function

Find the inverse of $h (x) = x^{3} + 2$ ‍.

h^{- 1} (x) =

3) Cube-root function

Find the inverse of $f (x) = 4 \cdot \sqrt[3]{x}$ ‍.

f^{- 1} (x) =

4) Rational functions

Find the inverse of $g (x) = \frac{x - 3}{x - 2}$ ‍.

g^{- 1} (x) =

To start, replace

g (x)

with

y

. Then solve for

x

\begin{aligned} g (x) & = \frac{x - 3}{x - 2} \\ y & = \frac{x - 3}{x - 2} & Replace g(x) with y \\ y (x - 2) & = x - 3 & Multiply both sides by x-2 \\ y x - 2 y & = x - 3 & Distribute \\ y x - x & = 2 y - 3 & Group terms with x on one side \\ x (y - 1) & = 2 y - 3 & Factor out an x \\ x & = \frac{2 y - 3}{y - 1} & Divide both sides by y-1 \\ g^{- 1} (y) & = \frac{2 y - 3}{y - 1} & Replace x with g^{- 1} (y) \end{aligned}

Since the choice of variable is arbitrary, we can now switch

y

x

to write the inverse in terms of

x

$g^{- 1} (x) = \frac{2 x - 3}{x - 1}$ ‍

5) Challenge problem

Match each function with the type of its inverse.

1

You can complete this problem without finding the actual inverses. Just think about what operations would "undo" the function. Or, use what was done above to help you.

Function	Inverse type
$f (x) = 3 x - 5$ ‍	Linear
$g (x) = x^{3} - 7$ ‍	Cube root
$h (x) = \frac{x + 2}{x - 3}$ ‍	Rational
$j (x) = \sqrt{x + 2}$ ‍	Quadratic

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20leunge
Posted 7 years ago. Direct link to 20leunge's post “Why are rational function...”
Why are rational functions called "rational"?
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(17 votes)
Answer
- Judith Gibson
  Posted 7 years ago. Direct link to Judith Gibson's post “Rational numbers are numb...”
  Rational numbers are numbers that can be expressed as a fraction (a ratio) of two integers.
  Rational functions are also fractions (ratios) - with polynomials in the numerator and denominator.
  Comment on Judith Gibson's post “Rational numbers are numb...”
  (28 votes)
Kennedy
Posted 7 years ago. Direct link to Kennedy's post “What is the inverse of th...”
What is the inverse of the function e to the power of x? Or some other exponential function?
Thank you in advance.
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(10 votes)
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alexasha2246
Posted 6 years ago. Direct link to alexasha2246's post “How do you find the inver...”
How do you find the inverse of y=x+3/3
?
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(5 votes)
Answer
- bears4347
  Posted 6 years ago. Direct link to bears4347's post “y=(x+3)/3 3y=x+3 3y-3=x ∴...”
  y=(x+3)/3
  3y=x+3
  3y-3=x
  ∴x=3y-3
  Button navigates to signup page
  (5 votes)
Xitlaly ortiz
Posted 6 years ago. Direct link to Xitlaly ortiz's post “how do you find the inver...”
how do you find the inverse of F(x)= x/(x-1)
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(2 votes)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “𝑦 = 𝑥/(𝑥 – 1) Swap the...”
  𝑦 = 𝑥/(𝑥 – 1)
  Swap the domain and range:
  𝑥 = 𝑦/(𝑦 – 1)
  Now solve for the inverse:
  𝑥𝑦 – 𝑥 = 𝑦
  𝑥𝑦 – 𝑦 = 𝑥
  𝑦(𝑥 – 1) = 𝑥
  𝑦 = 𝑥/(𝑥 – 1)
  So interestingly, this function is its own inverse. Functions like this are called involutions. Comment if you have questions!
  Comment on andrewp18's post “𝑦 = 𝑥/(𝑥 – 1) Swap the...”
  (10 votes)
andreamon3256
Posted 6 years ago. Direct link to andreamon3256's post “x-1 = -1/y+1, I got stuck...”
x-1 = -1/y+1, I got stuck on this part, I don't know how to get rid of a fraction with a denominator like that. help me please!
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- kubleeka
  Posted 6 years ago. Direct link to kubleeka's post “I assume y+1 is the denom...”
  I assume y+1 is the denominator, not just y?
  
  We take your equation and multiply both sides by y+1. On the right, we get -1(y+1)/(y+1), and the (y+1)'s cancel. So we're left with
  (x-1)(y+1)= -1
  Now we divide by (x+1), then subtract 1 to isolate y.
  Comment on kubleeka's post “I assume y+1 is the denom...”
  (2 votes)
Miles Woo
Posted 3 years ago. Direct link to Miles Woo's post “How do you find the inver...”
How do you find the inverse with a set of numbers?
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(3 votes)
Answer
juliannazhang
Posted 7 years ago. Direct link to juliannazhang's post “How would you find the in...”
How would you find the inverse of a function where x is greater/less than a specific number? Example problem: Write the inverse of f(x) = 2x+3 where x > 5
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(2 votes)
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Kalina Deleva
Posted 5 years ago. Direct link to Kalina Deleva's post “What is the inverse funct...”
What is the inverse function of f(x)= x*2 - x
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(3 votes)
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Arbaaz Ibrahim
Posted 5 years ago. Direct link to Arbaaz Ibrahim's post “On question number 4, I d...”
On question number 4, I don't understand why, we subtract x on from the right, instead of subtracting 'xy' from the left. We need to solve for 'x' right?
Can't solve it both ways?
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(3 votes)
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saleha iftikhar
Posted 5 years ago. Direct link to saleha iftikhar's post “this question is related ...”
this question is related to FUNCTIONS topic from maths
How to solve f^1(1/2)
when the only information given in the question is
f(x)=3/2x-5
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