Numeric Inverse Solver
Pattern Wizard
Get a closed form when the function matches a supported family.
What This Tool Does
Inverse functions map outputs back to inputs by reversing a one to one relation. To exist an inverse requires a function to be injective over the chosen domain. Restricting the domain can make a non injective rule invertible. For example the square rule is invertible on nonnegative numbers. The calculator focuses on identifying safe intervals and monotonic behavior. When monotonic the function has exactly one solution for each target value. That lets numeric solvers converge reliably to the unique preimage.
Supported Patterns and Strategy
This tool supports analytic patterns seen in study. Linear and affine rules allow direct algebraic inversion. Power rules and exponential rules are included alongside natural logarithm forms. Simple rational rules of the type ax plus b over cx plus d are supported when denominators stay nonzero. Trigonometric rules can be inverted on restricted domains. Where closed forms are not recognized the numeric engine takes over. Accurate inverses are found by solving f of x equals y within a safe interval.
Numerical Method Overview
The numeric engine uses bracketing and bisection to solve f of x equals a target value. Provide a lower bound and an upper bound where the function crosses the target within the interval. The method halves the interval until the width meets the tolerance. When the function is monotonic the result is unique. If the function oscillates shrink the interval to a monotonic region. Iteration and tolerance controls let you trade speed against precision. Diagnostics report residual and steps used.
Domain and Range Guidance
Choosing a correct domain and range is essential for invertibility. A function may be many to one across the real line yet become one to one on a restricted interval. Select an interval that avoids local extrema. Check monotonicity by sampling values at a grid of points and verifying consistent direction. If a turning point exists split the domain. The tool provides quick monotonic checks using finite differences. That guard helps bisection converge to a solution matching the desired branch.
How To Use This Calculator
Begin by entering a function of x using standard operations and supported functions. Enter the target y value whose preimage you seek. Supply a lower bound and an upper bound that bracket the solution. Pick a tolerance and iteration cap suited to your needs. Submit the form to compute the inverse value and see diagnostic details. Use the pattern wizard when a function fits a known family. That path yields a symbolic formula and evaluates it directly for the target.
Algebraic Inversion Basics
Algebraic inversion follows a simple philosophy. Rename the output as y. Solve the equation y equals f of x for x using legal operations that preserve equivalence. Swap roles so that x becomes the new input and y becomes the output. For linear rules the answer is y minus b over a. For exponential rules use natural logarithm. For rational rules apply cross multiplication and isolate x. When several branches exist state the domain restriction used to select the branch.
FAQs
Which functions are supported in the input?
Use basic operators with parentheses and these functions: sin cos tan asin acos atan sinh cosh tanh asinh acosh atanh sqrt log exp abs. Constants pi and e are available. The input must be a single valued function of x.
How do I choose bounds for the numeric method?
Pick a lower and an upper bound so that f(x) crosses the target y within the interval. If the solver says the target is not bracketed widen the interval or sample values to locate a sign change of f(x) minus y.
What does the monotonic check mean?
The checker samples points on the interval and tests whether values consistently increase or decrease. If the sign flips the function is not monotonic there and the inverse may not be unique. Try narrowing the interval to a region without turning points.
Can the tool return a symbolic inverse?
Yes when the Pattern Wizard matches the family of your function it will display a closed form and evaluate it for the chosen target value. Otherwise the numeric method provides a highly accurate result with diagnostics.
What if I see evaluation errors?
Evaluation fails when the expression has unknown tokens or goes outside the allowed domain of a function such as log of a non positive number. Check the expression and the bounds then try again.
Does the calculator handle multiple branches?
Quadratic and trigonometric families can have more than one branch. Use domain restrictions or branch controls in the wizard to select the one that matches your application.
How precise are the answers?
Precision depends on the tolerance and the smoothness of the function. The residual and final interval width provide a quantitative quality check. You can tighten the tolerance and raise the iteration cap for more digits when needed.