Are Parabolic Functions One to One?

No, parabolic functions are generally not one-to-one.

Understanding One-to-One Functions

A function is considered one-to-one (or injective) if every element in its range (output) corresponds to exactly one element in its domain (input). In simpler terms, no two distinct inputs produce the same output.

To visually determine if a function is one-to-one, we use the Horizontal Line Test:

• Horizontal Line Test: If any horizontal line drawn across the graph of a function intersects the graph at more than one point, then the function is not one-to-one. Conversely, if every horizontal line intersects the graph at at most one point, the function is one-to-one.

Why Parabolic Functions Fail the Horizontal Line Test

As explicitly stated in the reference: "No, a parabola is not a 1-1 function. It can be proved by the horizontal line test. Now, if we draw the horizontal lines, then it will intersect the parabola at two points in the graph."

Consider a common parabolic function, such as f(x) = x².

• Example:
  • If you input x = 2, the output is f(2) = 2² = 4.
  • If you input x = -2, the output is f(-2) = (-2)² = 4.

Here, two different inputs (2 and -2) result in the same output (4). When graphed, a horizontal line drawn at y = 4 would intersect the parabola at both x = 2 and x = -2, thus failing the Horizontal Line Test.

Parabolic functions, defined by quadratic equations like y = ax² + bx + c (where a ≠ 0), exhibit a characteristic symmetrical U-shape (opening upwards or downwards). This symmetry inherently means that for most y-values (all except the vertex), there will be two distinct x-values that produce that same y-value.

Can a Parabolic Function Be Made One-to-One?

While an entire parabolic function is not one-to-one, it can be made one-to-one by restricting its domain.

• Domain Restriction: By limiting the input values to only one side of the parabola's axis of symmetry, the function segment within that restricted domain will pass the Horizontal Line Test.
  • Example: For f(x) = x², if we restrict the domain to x ≥ 0 (only the right half of the parabola), then for every output, there is only one corresponding input. For instance, if f(x) = 4, then x must be 2 (as -2 is no longer in the allowed domain).

In summary, standard parabolic functions are not one-to-one due to their symmetrical nature, which allows multiple inputs to yield the same output, a characteristic easily identifiable through the Horizontal Line Test.