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

Algebra (all content)

Unit 7: Lesson 20

Finding inverse functions (Algebra 2 level)

Finding inverse functions

CCSS.Math:

HSF.BF.B.4

HSF.BF.B.4a

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 to b, then the inverse, f, start superscript, minus, 1, end superscript, must take b to a.

Or in other words, f, left parenthesis, a, right parenthesis, equals, b, \Longleftrightarrow, f, start superscript, minus, 1, end superscript, left parenthesis, b, right parenthesis, equals, 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, left parenthesis, x, right parenthesis, equals, 3, x, plus, 2.

Before we do that, let's first think about how we would find f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis.

To find f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, we need to find the input of f that corresponds to an output of 8. This is because if f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, x, then by definition of inverses, f, left parenthesis, x, right parenthesis, equals, 8.

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

So f, left parenthesis, 2, right parenthesis, equals, 8 which means that f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, 2

Finding inverse functions

We can generalize what we did above to find f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis for any y.

[Why did we use y here?]

To find f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, we can find the input of f that corresponds to an output of y. This is because if f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, x then by definition of inverses, f, left parenthesis, x, right parenthesis, equals, y.

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

So f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, start fraction, y, minus, 2, divided by, 3, end fraction.

Since the choice of the variable is arbitrary, we can write this as f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 2, divided by, 3, end fraction.

[Is there another way to do this?]

Check your understanding

1) Linear function

Find the inverse of g, left parenthesis, x, right parenthesis, equals, 2, x, minus, 5.

g, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

[I need help!]

2) Cubic function

Find the inverse of h, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2.

h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

[I need help!]

3) Cube-root function

Find the inverse of f, left parenthesis, x, right parenthesis, equals, 4, dot, cube root of, x, end cube root.

f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

[I need help!]

4) Rational functions

Find the inverse of g, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 3, divided by, x, minus, 2, end fraction.

g, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

[I need help!]

5) Challenge problem

Match each function with the type of its inverse.

[I need help!]

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20leunge
Posted 6 years ago. Direct link to 20leunge's post “Why are rational function...”
Why are rational functions called "rational"?
Button navigates to signup pageComment on 20leunge's post “Why are rational function...”
(16 votes)
Answer
- Judith Gibson
  Posted 6 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...”
  (26 votes)
Kennedy
Posted 6 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.
Button navigates to signup pageComment on Kennedy's post “What is the inverse of th...”
(9 votes)
Answer
alexasha2246
Posted 5 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 5 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 5 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)
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- andrewp18
  Posted 5 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...”
  (8 votes)
andreamon3256
Posted 5 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 5 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)
juliannazhang
Posted 6 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)
Answer
- Hassan
  Posted 3 years ago. Direct link to Hassan's post “That is just going to res...”
  That is just going to restrict the range of the function, which is the domain of the inverse function, but the inverse function's expression is going to be the same ( or at least in this example).
  In your example:
  x > 5 => 2x+5 > 13 let y=f(x) => y > 13
  
  which means that the domain of f inverse is all the real numbers greater than 13.
  Button navigates to signup page
  (1 vote)
Kalina Deleva
Posted 4 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)
Answer
- Hassan
  Posted 3 years ago. Direct link to Hassan's post “`f(x)=x*2 - x` can be sim...”
  f(x)=x*2 - x can be simplified to f(x)=x.
  in which case the inverse of f seems to be itself.
  Button navigates to signup page
  (1 vote)
Arbaaz Ibrahim
Posted 4 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)
Answer
saleha iftikhar
Posted 4 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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(2 votes)
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Miles Woo
Posted 2 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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(2 votes)
Answer