Find an Inverse Function Formula and Check Its Domain

Check whether the function can be reversed uniquely

An inverse function assigns each output back to exactly one original input. A function must therefore be one-to-one on the domain being considered. If two different inputs produce the same output, reversing the relation would give two possible outputs for one inverse input. The function x² over all real numbers has that problem because 2 and −2 both produce 4. Restricting its original domain to nonnegative inputs makes a single-valued square-root inverse possible.

Rearrange a linear function step by step

For f(x)=3 x−4, write y=3 x−4. Add 4 to obtain y+4=3 x, then divide by 3 to obtain x=(y+4)/3. Renaming the inverse input as x gives f^−1(x)=(x+4)/3. The original process multiplies by 3 and subtracts 4; the inverse reverses those operations by adding 4 and dividing by 3. Reversing their order is essential. The expression x/3+4 would describe a different function and would fail a composition check.

Verify two input-output pairs numerically

The original function sends 5 to 11, so its inverse must send 11 back to 5. In the local Algebra Calculator, solve 3*x-4=11 and check the solution x=5. For an independent pair, the original sends 2 to 2, so solve 3*x-4=2 and check x=2. These point checks support the arithmetic but do not establish a formula for every possible input. The algebraic rearrangement and composition identities provide that broader justification.

Check both compositions

Substituting the proposed inverse into the original gives f(f^−1(x))=3((x+4)/3)−4=x. In the other direction, f^−1(f(x))=((3 x−4)+4)/3=x. Both identities hold for every real x in this linear example. In a restricted-domain problem, each composition must be checked on the appropriate domain. A formula that simplifies algebraically can still violate a required input restriction, so include domain statements rather than treating cancellation as the whole proof.

Find an inverse for a restricted quadratic

Let f(x)=(x−2)² with original domain x≥2. Its range is y≥0. Taking the nonnegative square root of y gives x−2=√y, so the inverse is f^−1(x)=2+√x for x≥0. If the original domain were x≤2, the inverse would instead be 2−√x. Without a domain restriction, the quadratic is not one-to-one and has no inverse function over its full real domain. Choosing a plus sign automatically would hide that important condition.

Track excluded values in a rational example

For f(x)=(2 x+3)/(x−1), the original domain excludes 1. Solve y(x−1)=2 x+3 to obtain x(y−2)=y+3 and x=(y+3)/(y−2). Thus f^−1(x)=(x+3)/(x−2), with inverse domain x≠2. The original output cannot equal 2, which explains that inverse restriction. Domain and range exchange roles: the inverse's range excludes 1. These restrictions should accompany the formula because they describe where the algebraic divisions are valid.

Distinguish inverse notation from a reciprocal

The notation f^−1(x) conventionally means the inverse function, not 1/f(x). For f(x)=3 x−4, the reciprocal is 1/(3 x−4), whereas the inverse is (x+4)/3. At x=11, the reciprocal is 1/29 but the inverse is 5. A matrix inverse uses the same superscript notation in a different algebraic setting, so read what object is being inverted. Selecting a matrix calculator because an exercise says inverse would not solve this function-rearrangement task.

Interpret the graph and the final domain

The graph of an inverse function is the reflection of the original graph across y=x. Thus a point (5,11)on f becomes(11,5)on its inverse. This geometric check agrees with the input-output reversal, but plotting alone does not replace the one-to-one condition or domain analysis. Report the inverse formula together with its valid inputs, and identify any original restriction used to obtain a single branch. If the function cannot be reversed uniquely on the specified domain, explain that before proposing a restricted version.

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