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## Algebra (all content)

### Unit 7: Lesson 20

Finding inverse functions (Algebra 2 level)

# Finding inverse functions

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.
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.

## 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

### 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

### 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

### 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

### 5) Challenge problem

Match each function with the type of its inverse.

## Want to join the conversation?

• Why are rational functions called "rational"?
(16 votes)
• 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.
(26 votes)
• What is the inverse of the function e to the power of x? Or some other exponential function?
Thank you in advance.
(9 votes)
• How do you find the inverse of y=x+3/3
?
(5 votes)
• y=(x+3)/3
3y=x+3
3y-3=x
∴x=3y-3
(5 votes)
• how do you find the inverse of F(x)= x/(x-1)
(2 votes)
• 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!
(8 votes)
• 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!
(3 votes)
• 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.
(2 votes)
• 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
(2 votes)
• 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 > 13let y=f(x)=> y > 13

which means that the domain of f inverse is all the real numbers greater than 13.
(1 vote)
• What is the inverse function of f(x)= x*2 - x
(3 votes)
• f(x)=x*2 - x can be simplified to f(x)=x.
in which case the inverse of f seems to be itself.
(1 vote)
• 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?
(3 votes)
• 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
(2 votes)
• How do you find the inverse with a set of numbers?
(2 votes)