Question

is the inverse of the function $f(x)=3x^2 - 2x + 1$ a function, and why or why not?

Explanation:

Step1: Identify function type

The function \(f(x) = 3x^2 - 2x + 1\) is a quadratic (parabola).

Step2: Apply Horizontal Line Test

A parabola opens upward/downward, so it fails the Horizontal Line Test: there exist horizontal lines that intersect the graph at 2 distinct points.

Step3: Relate to inverse definition

For a relation to be a function, each input must map to exactly one output. Since the original function has multiple \(x\)-values mapping to the same \(y\)-value, its inverse will have a single \(x\)-value mapping to multiple \(y\)-values, violating the function definition.

Answer:

Yes, the inverse of \(f(x) = 3x^2 - 2x + 1\) is not a function.