Confusing notation $D(p)(x)$ in a vector space of polynomials

Question

If we have a vector space that consists of all polynomials of degree less than or equal to 4, and we consider the following function: $$D(p)(x) = 2.5\cdot p(x-1)$$ where $p$ is a function from the above mentioned vector space..... what is meant my "$D(p)(x)$". I don't get the notation at all. Is $p$ an input, or $x$, or both at the same time? How does one show linearity, for example? Do I then just look at $(p+q)(x)$, two functions, or also $x + y$, two numbers?

Answer 1

The letter $p$ denotes a polynomial of degree less than or equal to $4$ (i.e. an element of the vector space of interest). If $D$ is an operator from the vector space to itself, then $D(p)$ must also be an element of the vector space, i.e. a polynomial of degree less than or equal to $4$. As $D(p)$ is a polynomial, we can consider its value at a point $x$ in its domain. According to the definition, the value of the polynomial $D(p)$ at the point $x$ is $2.5p(x-1)$, and we denote this $D(p)(x) = 2.5p(x-1)$.

To address the question of linearity, you need to show that $D(p+q) = D(p) + D(q)$. Note that a function is completely determined by its domain and its values. Any two polynomials have the same domain, so to show two polynomials are equal, it is enough to demonstrate that they have the same values. That is, in order to show that the polynomial $D(p+q)$ is equal to the polynomial $D(p) + D(q)$, one can show that $D(p+q)(x) = (D(p) + D(q))(x)$ for every $x$. Note, the latter simplifies to $D(p)(x) + D(q)(x)$ by the definition of function addition.