Inverse Function Verifier
Easily check if two functions are inverses of each other! Just enter f(x) and a potential inverse g(x) to see if they satisfy the inverse function conditions: $$f(g(x)) = x$$ and $$g(f(x)) = x$$.
Enter the first function in terms of x.
Enter the function you suspect is the inverse of f(x).
Verification Result
Is g(x) the inverse of f(x)?
Function Composition Analysis
Analysis of f(g(x))
For g(x) to be the inverse of f(x), the composition $$f(g(x))$$ must simplify to $$x$$.
Observation: f(g(x)) does not simplify to $$x$$.
Observation: f(g(x)) simplifies to $$x$$.
Analysis of g(f(x))
Similarly, for g(x) to be the inverse of f(x), the composition $$g(f(x))$$ must also simplify to $$x$$.
Observation: g(f(x)) does not simplify to $$x$$.
Observation: g(f(x)) simplifies to $$x$$.
Conclusion
Congratulations! 🎉 Both f(g(x)) and g(f(x)) simplify to $$x$$. Therefore, g(x) is indeed the inverse of f(x).
Unfortunately, either f(g(x)) or g(f(x)) (or both) do not simplify to $$x$$. Therefore, g(x) is not the inverse of f(x).
Understanding Inverse Functions
An inverse function reverses the operation of another function. If f(x) produces y, then the inverse function, denoted as f⁻¹(y), produces x. For g(x) to be the inverse of f(x), two conditions must be met:
- $$f(g(x)) = x$$: Applying g first, then f, returns the original input x.
- $$g(f(x)) = x$$: Applying f first, then g, also returns the original input x.
This tool verifies these conditions by simplifying the compositions $$f(g(x))$$ and $$g(f(x))$$. If both simplify to x, then g(x) is the inverse of f(x).
Further learning: Wikipedia, Khan Academy.