What is the inverse function of f(x)=x^4+27x-6?
- PopeLv 74 weeks ago
People generally do not believe me when I say this, but here goes one more try. Not all functions have an inverse. The function you gave is a polynomial of even degree, and there are no constraints on the domain. As a result, it is not injective, and cannot have an inverse. The injectivity requirement is not some quibble over a minor point. It is a fundamental property of inverse functions.
Here is proof by contradiction. Suppose function f has an inverse, f⁻¹.
f(x) = x⁴ + 27x - 6
The function, being a polynomial, has the domain of all real numbers.
For any real x, f⁻¹[f(x)] = x.
f(-3) = -6
f(0) = -6
-3 = f⁻¹[f(-3)] = f⁻¹(-6) = f⁻¹[f(0)] = 0
-3 = 0
However, -3 is not equal to 0. By contradiction, we must reject the proposition that function f has an inverse.
There legions of evil mathematics teachers out there wishing to tell you otherwise. Some say that all functions have inverses, but that sometimes the inverse itself is not a function. That flies in the face of the definition, and contradicts the f⁻¹[f(x)] condition shown above. Others will pointlessly look for a domain on which the function would be injective. That might be possible, but to change the domain of a function is to change its definition, and that in turn would be changing the question to fit the answer.