Show Algebraically Inverse Functions Calculator

Calculator

Function Type

Coefficient a

Coefficient b or Base b

Coefficient c

Coefficient d

Horizontal Shift h

Vertical Shift k

Power n

Target Output y

Branch for Even Powers

Positive branch

Negative branch

Example Data Table

Function Type	Inputs	Original Function	Inverse Form	Sample Target
Linear	a = 2, b = 3	f(x) = 2x + 3	f^-1(x) = (x - 3) / 2	11
Quadratic Branch	a = 1, h = 2, k = 5	f(x) = (x - 2)^2 + 5	f^-1(x) = 2 + sqrt(x - 5)	14
Exponential	a = 3, b = 2, h = 1, k = 4	f(x) = 3 * 2^(x - 1) + 4	f^-1(x) = 1 + log((x - 4) / 3) / log(2)	28
Fractional Linear	a = 2, b = 3, c = 1, d = 4	f(x) = (2x + 3) / (x + 4)	f^-1(x) = (3 - 4x) / (x - 2)	1.4

Formula Used

The calculator isolates x from y = f(x). Then it swaps x and y to create f^-1(x).

Linear: if y = ax + b, then f^-1(x) = (x - b) / a.
Quadratic branch: if y = a(x - h)^2 + k, then f^-1(x) = h ± sqrt((x - k) / a).
Cubic: if y = a(x - h)^3 + k, then f^-1(x) = h + cbrt((x - k) / a).
Power: if y = a(x - h)^n + k, then f^-1(x) = h + root_n((x - k) / a), with branch rules for even n.
Exponential: if y = ab^(x - h) + k, then f^-1(x) = h + log((x - k) / a) / log(b).
Logarithmic: if y = a log_b(x - h) + k, then f^-1(x) = h + b^((x - k) / a).
Reciprocal: if y = a / (x - h) + k, then f^-1(x) = h + a / (x - k).
Fractional linear: if y = (ax + b) / (cx + d), then f^-1(x) = (b - dx) / (cx - a).

How to Use This Calculator

Select the function family that matches your algebra problem.
Enter the needed coefficients. Unused fields can stay unchanged.
Use coefficient b as the base for exponential and logarithmic functions.
Choose a branch for quadratics and even powers.
Enter the target output y that you want to reverse.
Press the calculate button to see the inverse result above the form.
Review the algebraic steps and verification value.
Download the result as CSV or PDF when needed.

Algebraic Inverse Functions Guide

Why Inverses Matter

Inverse functions help you move from an output back to its matching input. This calculator focuses on algebraic reversal. It does not only give a final expression. It also shows the key transformations used to isolate x.

Supported Function Models

The tool supports common families used in algebra, precalculus, and applied math. You can solve linear, shifted power, cubic, exponential, logarithmic, reciprocal, and fractional linear functions. Each model has fields for its coefficients. The calculator then builds the original function, applies inverse rules, and evaluates a chosen output value.

Branches, Domains, and Ranges

An inverse exists on a domain where the original function is one to one. That point matters for quadratics and even powers. These functions normally fail the horizontal line test over all real numbers. A branch restriction fixes that issue. Choose the positive branch for inputs to the right of the shift. Choose the negative branch for inputs to the left. The displayed domain and range notes help you read that condition clearly.

Checking the Answer

The calculated inverse value answers a direct question. Which input produces this target output? For example, if f(x) = 3x + 6 and y = 21, the inverse returns x = 5. The check value then puts 5 back into the original function. A matching result confirms the algebra.

Learning and Records

This page is useful for learning and record keeping. Students can compare symbolic steps with numeric answers. Teachers can make examples quickly. Analysts can reverse simple transformation formulas without writing notes by hand.

Export and Validation

The export buttons support clean documentation. Use CSV for spreadsheet records. Use PDF for a readable report. The example table shows typical entries before you submit your own values. For best results, enter nonzero scale values where required. Avoid invalid bases for exponential and logarithmic forms. Also watch for restricted outputs, such as values outside a square root range. Because the work is shown in steps, the calculator also supports error checking. If a denominator becomes zero, the report explains the restriction. If a target output cannot be reversed, it marks the value as invalid instead of hiding the issue. This makes the result safer for homework review, lesson planning, and model testing. You can change one coefficient at a time and see how the inverse expression shifts. This improves pattern sense.

FAQs

What does this calculator do?

It shows the algebraic inverse of supported function types. It also evaluates the inverse at a target output value and checks the answer in the original function.

Why do some functions need a branch?

Quadratics and even powers are not one to one over all real numbers. A branch limits the domain, so the inverse becomes a proper function.

Can I use decimals?

Yes. The input fields accept decimal values. The result is rounded for clean display, but the calculation uses floating point arithmetic internally.

What is the target output y?

It is the output value you want to reverse. The calculator finds the input x that would produce that output in the original function.

Why is my result invalid?

The target may violate a domain or range rule. Examples include a negative square root input, zero denominator, or invalid logarithm input.

Which field is the exponential base?

Use the b field as the base for exponential and logarithmic functions. The base must be positive and cannot equal one.

Does this prove two functions are inverses?

It shows the inverse construction for a selected model. To prove two separate functions are inverses, also check both compositions.

What does the CSV export include?

The CSV file includes the selected model, formula, inverse form, target output, inverse value, domain, range, status, and algebraic steps.