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Inverse Function Solver

Compute f^{-1}(x) by swapping x and y and solving for y

Select Function Type

Linear ax+b Quadratic ax^2+c Exponential a^x

f(x)=ax+b

f(x)=x+

Inverse Function Method

1. Replace f(x) with y

2. Swap x and y variables

3. Solve for y

4. Write f^{-1}(x) = y

The inverse function undoes the original function. Finding the inverse requires swapping x and y and solving for y. The graph of f^{-1} is the reflection of f across the line y=x. Only one-to-one functions have inverses.

⚠ Not all functions have inverses. Only one-to-one (horizontal line test passes) functions have inverses. Quadratics need domain restrictions.

What Is an Inverse Function?

The inverse function f^{-1} satisfies f(f^{-1}(x))=x and f^{-1}(f(x))=x. It reverses the mapping of f. To find it: swap x and y, then solve for y. The domain of f becomes the range of f^{-1} and vice versa.

Linear Inverse

f(x)=ax+b. Inverse: f^{-1}(x)=(x-b)/a. Easy to compute. Always exists if a!=0.

Quadratic Inverse

f(x)=ax^2+c (x>=0). Inverse: f^{-1}(x)=sqrt((x-c)/a). Needs domain restriction for one-to-one.

Exponential Inverse

f(x)=a^x. Inverse is log: f^{-1}(x)=log_a(x). The log base a of x. Domain x>0.

Verification

Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Both must hold. Graphically: symmetric about y=x.

Teaching Example: f(x)=2x-3. y=2x-3 -> swap: x=2y-3 -> solve: 2y=x+3 -> y=(x+3)/2. f^{-1}(x)=(x+3)/2. Verify: f(f^{-1}(x))=2*(x+3)/2-3=x+3-3=x.

Applications

Algebra Calculus Cryptography Physics Engineering

Frequently Asked Questions

What is inverse function?▼

f^{-1} undoes f. Swap x and y, solve for y. Reflection across y=x. Only for one-to-one functions.

How to find inverse?▼

Replace f(x) with y. Swap x,y. Solve for y. Write f^{-1}(x)=result. Verify with composition.

Linear always invertible?▼

Linear functions with a!=0 always have inverses. f(x)=ax+b -> f^{-1}(x)=(x-b)/a. Only if slope is nonzero.

Exponential inverse?▼

f(x)=a^x -> f^{-1}(x)=log_a(x). The logarithm (base a) is the inverse of exponentiation.

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