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An injective function, also known as a one-to-one function, is a type of function with a unique characteristic: no two different elements (inputs) in the domain (set of all possible inputs) will produce the same element (output) in the codomain (set of all possible outputs).

In simpler terms, imagine you have two distinct keys, and each key can open only one unique lock. That’s how injective functions work. Every input has a unique output. For a function, say f(x), to be injective, whenever we have f(a) = f(b), it must follow that a = b.

However, if we encounter a situation where f(a) = f(b) but a ≠ b, the function is not injective. Taking the example provided, for the function f(x) = x^2 + 1, we found that f(3) = f(-3); since 3 and -3 are distinct numbers, this function is not one-to-one or injective, because it assigns the same output (10) to multiple inputs.

A quadratic function is a polynomial function of the second degree, meaning its highest power of x is 2. The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. Quadratic functions graph as parabolas, which are U-shaped curves that can open upwards or downwards depending on the sign of a.

In the same exercise, f(x) = x^2 + 1 is a quadratic function since it fits the form with a = 1, b = 0, and c = 1. This particular quadratic function will graph a parabola opening upwards (because a is positive) and shifted up the y-axis by 1 (since c = 1). A distinctive property of quadratic functions is their symmetry along a vertical line known as the axis of symmetry, which, in this case, is the y-axis since the function is symmetrical about x = 0. This symmetry also explains why the outputs for f(3) and f(-3) are the same.

The process of function evaluation involves substituting a specific value into the function for the variable x and calculating the result. It's like following a recipe where the input is analogous to the ingredients and the function's formula is the cooking procedure; you end up with a dish (output), which varies depending on the ingredients (input) you started with.

To evaluate the function f(x) at x = 3, you would replace every instance of x in the function's formula with 3. This was demonstrated in the exercise where f(x) = x^2 + 1 was evaluated for x = 3 and x = -3, yielding the results f(3) = 10 and f(-3) = 10 respectively.

It's important when evaluating functions to pay close attention and perform each step in the correct order, typically following the order of operations (PEMDAS/BODMAS). This ensures accurate results and helps identify the properties of the function, such as whether it is injective or not.