Inverse Functions

Learn how to find and verify inverse functions, and understand their relationship to the original function.

Advanced25 minLesson
Printable worksheet

Definition

An inverse function reverses the action of the original function. If f(x)f(x) takes input aa and produces output bb, then the inverse function f−1(x)f^{-1}(x) takes input bb and produces output aa.
If f(a)=b, then f−1(b)=a\text{If } f(a) = b, \text{ then } f^{-1}(b) = a
Key Properties:
  • The inverse "undoes" what the original function does
  • f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x
  • The graph of f−1(x)f^{-1}(x) is the reflection of f(x)f(x) over the line y=xy = x
  • Not all functions have inverses - only one-to-one functions do
Finding the Inverse:
  1. 1.Replace f(x)f(x) with yy
  2. 2.Swap xx and yy
  3. 3.Solve for yy
  4. 4.Replace yy with f−1(x)f^{-1}(x)

Try it now

If f(3)=7f(3) = 7, what is f−1(7)f^{-1}(7)?

Worked Examples

Find the inverse of f(x)=2x+3f(x) = 2x + 3
1

Replace f(x) with y

y=2x+3y = 2x + 3 → Standard form
2

Swap x and y

x=2y+3x = 2y + 3 → Variables swapped
3

Solve for y: subtract 3

x−3=2yx - 3 = 2y → Isolate the y term
4

Divide by 2

y=x−32y = \frac{x - 3}{2} → y is isolated
5

Write as inverse function

f−1(x)=x−32f^{-1}(x) = \frac{x - 3}{2} → Inverse found

Common Mistakes

Confusing f−1(x)f^{-1}(x) with 1f(x)\frac{1}{f(x)}

Why it's wrong: The notation f−1f^{-1} means the inverse function, NOT the reciprocal. f−1(x)f^{-1}(x) undoes f(x)f(x), while 1f(x)\frac{1}{f(x)} is just one divided by f(x)f(x).

Correct: f−1(x)f^{-1}(x) is the inverse function. For the reciprocal, write 1f(x)\frac{1}{f(x)} or (f(x))−1(f(x))^{-1}.

Forgetting to swap x and y before solving

Why it's wrong: The swap is essential because the inverse reverses inputs and outputs. Without swapping, you'll just solve for x in terms of y, not find the inverse.

Correct: Always follow the steps: write y = f(x), swap to get x = f(y), then solve for y.

Assuming all functions have inverses

Why it's wrong: Only one-to-one functions (passing the horizontal line test) have inverses. Functions like f(x)=x2f(x) = x^2 fail because two inputs give the same output.

Correct: Check that the function is one-to-one first. For f(x)=x2f(x) = x^2, restrict the domain to x≥0x \geq 0 to create an inverse.

Forgetting domain restrictions for the inverse

Why it's wrong: The domain of f−1f^{-1} is the range of ff, and vice versa. If the original function excludes certain values, the inverse will have different restrictions.

Correct: Always state the domain of the inverse. If f(x)=1x−2f(x) = \frac{1}{x-2} has domain x≠2x \neq 2, then f−1f^{-1} will have a different restriction.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Inverse Functions Explained | Table, Graph & Function Examples

by Understand The Math

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How to Find the Inverse of a Function | Step-by-Step Examples

by Understand The Math

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Try these examples:

y = 2x + 3

m=2, b=3

y = (x - 3)/2

m=0.5, b=-1.5

y = x

m=1, b=0

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Practice Problems

18 problems
Problem 1 of 18
Easy
If f(3)=7f(3) = 7, what is f−1(7)f^{-1}(7)?

Why It Matters

Inverse functions are essential in many real-world applications:
  • Cryptography: Encryption functions need inverses for decryption
  • Unit Conversion: Converting Celsius to Fahrenheit and back requires inverse functions
  • Finance: Calculating principal from compound interest requires inverting the growth formula
  • Science: Finding time from distance (inverse of position function)
  • Computer Graphics: Undoing transformations requires inverse operations
Understanding inverses is crucial for solving equations and forms the foundation for logarithms (inverse of exponentials) and inverse trigonometric functions.

Real World Applications

Temperature Conversion

Converting between Celsius and Fahrenheit uses inverse functions.

Example:

The formula F=95C+32F = \frac{9}{5}C + 32 converts Celsius to Fahrenheit. Its inverse C=59(F−32)C = \frac{5}{9}(F - 32) converts back.

1Try It Yourself

You know the Fahrenheit to Celsius formula is C=59(F−32)C = \frac{5}{9}(F - 32).

Find the inverse function to convert Celsius back to Fahrenheit.

Step 1: Write the mathematical expression

Swap variables and solve for F:

Decryption in Cryptography

Simple encryption functions need inverses for decryption. In a Caesar cipher, if encryption shifts letters by 3, decryption shifts by -3.

Example:

If encryption is E(x)=x+3(mod26)E(x) = x + 3 \pmod{26}, decryption is D(x)=x−3(mod26)D(x) = x - 3 \pmod{26}

2Try It Yourself

A simple encryption function is E(x)=2x+5E(x) = 2x + 5.

Find the decryption function (inverse).

Step 1: Write the mathematical expression

Find E−1(x)E^{-1}(x):

Key Takeaways

  • 1An inverse function f−1(x)f^{-1}(x) reverses the action of f(x)f(x): if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a
  • 2To find an inverse: replace f(x)f(x) with yy, swap xx and yy, solve for yy, write as f−1(x)f^{-1}(x)
  • 3To verify inverses: check that f(f−1(x))=xf(f^{-1}(x)) = x AND f−1(f(x))=xf^{-1}(f(x)) = x
  • 4Only one-to-one functions (passing the horizontal line test) have inverses
  • 5The graph of f−1(x)f^{-1}(x) is the reflection of f(x)f(x) over the line y=xy = x
  • 6The domain of f−1f^{-1} equals the range of ff, and the range of f−1f^{-1} equals the domain of ff

Frequently Asked Questions

f−1(x)f^{-1}(x) is the inverse function - it undoes ff. f(x)−1f(x)^{-1} or (f(x))−1(f(x))^{-1} is the reciprocal 1f(x)\frac{1}{f(x)}. They are completely different! For example, if f(x)=2xf(x) = 2x, then f−1(x)=x2f^{-1}(x) = \frac{x}{2}, but f(x)−1=12xf(x)^{-1} = \frac{1}{2x}.
f−1(x)f^{-1}(x) is the inverse function - it undoes ff. f(x)−1f(x)^{-1} or (f(x))−1(f(x))^{-1} is the reciprocal 1f(x)\frac{1}{f(x)}. They are completely different! For example, if f(x)=2xf(x) = 2x, then f−1(x)=x2f^{-1}(x) = \frac{x}{2}, but f(x)−1=12xf(x)^{-1} = \frac{1}{2x}.
A function has an inverse if and only if it is one-to-one (each output comes from exactly one input). Use the horizontal line test: if any horizontal line crosses the graph more than once, the function does NOT have an inverse.
Because the inverse swaps inputs and outputs. If ff takes 2 and gives 7, then f−1f^{-1} takes 7 and gives 2. Swapping x and y in the equation captures this reversal of roles.

Glossary

Inverse function
A function f−1(x)f^{-1}(x) that reverses the action of f(x)f(x), so that f−1(f(x))=xf^{-1}(f(x)) = x
One-to-one function
A function where each output value corresponds to exactly one input value
Horizontal line test
If any horizontal line intersects the graph more than once, the function is not one-to-one and has no inverse
Composition
Applying one function to the result of another, written as f(g(x))f(g(x)) or (f∘g)(x)(f \circ g)(x)
Domain
The set of all valid input values for a function
Range
The set of all possible output values of a function

Formula Card

Inverse Definition

f(a)=b⇔f−1(b)=af(a) = b \Leftrightarrow f^{-1}(b) = a

If f maps a to b, then f inverse maps b back to a

Composition Identity

f(f−1(x))=f−1(f(x))=xf(f^{-1}(x)) = f^{-1}(f(x)) = x

Composing a function with its inverse gives x

Domain-Range Relationship

Domain of f−1=Range of f\text{Domain of } f^{-1} = \text{Range of } f

Domains and ranges swap for inverses

Today's challengeOne problem, three tries

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