High School Inverse Function Finder
Select a function type to find its inverse f⁻¹(x) and visualize the symmetry over y = x.
Inverse Equation f⁻¹(x)
Algebraic Steps
Graph Visualization
How this Inverse Function Finder works
In mathematics, an inverse function reverses the operation of the original function. If a function f takes an input x to an output y, then the inverse function f⁻¹ takes that y back to x. Symbolically, f(f⁻¹(x)) = x.
Algebraic Method
To find the inverse algebraically, we follow a standard four-step process:
- Replace f(x) with y.
- Swap x and y in the equation. This step reflects the geometric property of reflecting across the line y = x.
- Solve the new equation for y.
- Replace y with f⁻¹(x).
One-to-One Functions
Not all functions have inverses that are also functions. For a function to have an inverse, it must be “one-to-one,” meaning it passes the Horizontal Line Test. No horizontal line should intersect the graph more than once. For example, a standard parabola y = x² fails this test (it hits -2 and 2 for y=4), so it does not have a true inverse function unless we restrict its domain (usually to x ≥ 0).
Why use an Inverse Function Finder?
Visualizing the relationship between a function and its inverse is crucial for understanding concepts like domain, range, and asymptotes. The graph clearly shows how the domain of f(x) becomes the range of f⁻¹(x), and vice versa.
FAQ
Yes, this tool is completely free to use for educational purposes.
Inverse functions are reflections of each other across the line y = x. This is because we literally swap the x and y coordinates.
Currently, this tool focuses on Linear and Rational functions, which are always one-to-one (within their domains) and invertible without restricting domains.
It is a ratio of polynomials, like a fraction with x in the denominator. Finding their inverse involves factoring and grouping terms.