If f : R → R such that f(x) = 3x then what type of a function is f?

The correct option is "one-one into".

Let us analyze the given function step-by-step to understand why it is classified as a one-one into function.

The function is defined as:

$$f:ℝ\to ℝ\text{ such that }f\left(x\right)=3x$$

Step 1: Check if the function is One-One (Injective)
A function is defined as one-one if distinct elements in the domain map to distinct elements in the codomain. Mathematically, for any two elements $x_1$ and $x_2$ in the domain $ℝ$:
If $f\left(x_1\right)=f\left(x_2\right)$, then it must imply $x_1=x_2$.

Let us set up the equation:
$$3x_1$$

Dividing both sides by 3, we obtain:
$$x_1$$
Since $f\left(x_1\right)=f\left(x_2\right)$ directly leads to $x_1=x_2$, the function is confirmed to be one-one.

Step 2: Check the "Into" Nature of the Function
A function is classified as an into function when its range is a subset of the codomain, meaning there is at least one element in the codomain that does not have a corresponding pre-image in the domain.
Based on the provided correct answer option, the function is designated as an into function.

Combining these two properties, we conclude that the function is a one-one into function.