Enter function type to automatically swap x and y and find f⁻¹(x)
Select Function Type
Enter linear function parameters (y = ax + b)
y =x +
Enter square root parameters (y = √(x) + c)
y = √(x) +
Result
Original Function
Inverse Function
Step-by-Step Derivation
Inverse Function Steps
1. Swap x and y in y=f(x) to get x=f(y)
2. Solve the new equation x=f(y) for y
3. Replace y with f⁻¹(x)
The inverse function reflects a "reverse mapping" relationship. If the original describes how input maps to output, the inverse describes how to recover the original input from the output.
⚠Only monotonic (one-to-one) functions have inverses. y=x² has no inverse on R, but with domain restriction x≥0, its inverse is y=√x. The domain of the inverse equals the range of the original, and vice versa.
What Is an Inverse Function?
An inverse function reverses the original function. If y = f(x) maps x to y, then its inverse f⁻¹(x) maps y back to x, satisfying f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The graphs of a function and its inverse are symmetric about the line y=x.
Definition
If y = f(x) maps x to y, the inverse f⁻¹(x) maps y back to x, achieving reverse correspondence.
Solution Steps
Swap x and y in y=f(x) to get x=f(y), solve the new equation for y, then replace y with f⁻¹(x).
Condition
A function has an inverse iff it is strictly monotonic (one-to-one) on its domain. Non-monotonic functions need domain restriction.
Graph Relationship
The graphs are symmetric about y=x. The original's domain is the inverse's range, and vice versa.
Teaching Example: Find the inverse of y = 2x + 3.
1. Swap x and y: x = 2y + 3
2. Solve for y: 2y = x - 3, y = (x - 3) / 2
3. Inverse: f⁻¹(x) = (x - 3) / 2
Applications
High School MathInverse RelationsExam PrepCompetition MathPrecalculus
Frequently Asked Questions
What is an inverse function?▼
An inverse function reverses the original function. If y = f(x), the inverse is denoted as x = f⁻¹(y). It satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
How do you find the inverse of a function?▼
1. Swap x and y in y=f(x) to get x=f(y); 2. Solve the new equation for y; 3. Replace y with f⁻¹(x). The original's domain and range become the inverse's range and domain respectively.
Do all functions have inverses?▼
No. Only strictly monotonic (one-to-one) functions have inverses on their entire domain. For example, y=x² on R has no inverse, but restricting x≥0 gives the inverse y=√x.
What is the inverse of a linear function?▼
For y=ax+b (a≠0), the inverse is y=(x-b)/a. Swap x and y: x=ay+b → y=(x-b)/a. For example, the inverse of y=2x+3 is y=(x-3)/2.
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