About identification of type of functions: injective, surjective, or bijective. Also, identification of into functions.

Steps to Solve

1. Identify the Definitions of Functions

Understand the basic definitions:

• An injective function (or one-to-one function) is a function where different inputs lead to different outputs. Formally, if $f(a) = f(b)$ implies that $a = b$, then $f$ is injective.

• A surjective function (or onto function) is a function where every element in the codomain is the image of at least one element from the domain. Formally, for every $y$ in the codomain $Y$, there exists at least one $x$ in the domain $X$ such that $f(x) = y$.

• A bijective function is a function that is both injective and surjective. This means it is a one-to-one mapping between the domain and the codomain.

2. Understand 'Into' Functions

'Into' functions refer to functions that are not surjective. This means that not every element of the codomain is covered by the function. Thus, there exists at least one element in the codomain that is not an image of any element in the domain.

3. Analyze Examples

Consider examples that illustrate each type of function:

• Injective Example: The function $f(x) = 2x$ is injective because different values of $x$ lead to different values of $f(x)$.

• Surjective Example: The function $f(x) = x^3$ is surjective when considered from $\mathbb{R}$ to $\mathbb{R}$ because every real number $y$ can be represented as $y = x^3$ for some real $x$.

• Bijective Example: The function $f(x) = x + 1$ is bijective from $\mathbb{R}$ to $\mathbb{R}$ because it is both injective and surjective.

• Into Function Example: The function $f(x) = x^2$ is into when defined from $\mathbb{R}$ to $\mathbb{R}$, as it cannot produce negative values in its range.