Compute f^{-1}(x) by swapping x and y and solving for y
Select Function Type
f(x)=ax+b
f(x)=x+
f(x)=ax^2+c, x>=0
f(x)=x^2+
f(x)=a^x
f(x)=^x
Result
Derivation
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.
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