Is this $f:R^2\to R$ with $x_1, x_2\mapsto \beta _{0} + \beta_{1}x_{1} + \beta_{2}x_{2}$ the right form of the definition of a function?

Question

Per wiki

In statistics, linear regression is a linear approach to modeling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent variables).

the model takes the form

${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta > _{p}x_{ip}+\varepsilon _{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i},\qquad i=1,\ldots ,n,}$

where T denotes the transpose, so that $x_i^Tβ$ is the inner product between vectors $x_i$ and β.

simplified,

$y_1 = \beta _{0} + \beta_{1}x_{1} + \beta_{2}x_{2} \tag{1}$

I guess equation 1 could be represented as a function, things like

$f:R^2\to R$ with $x_1, x_2\mapsto \beta _{0} + \beta_{1}x_{1} + \beta_{2}x_{2}$

So, is above the right form of the definition of a function?

Answer 1

Using ordered pairs is normal for such functions. You may write $$f(x_1,x_2)=\alpha_0 +\alpha_1x_1+\alpha_2x_2$$ or $$f:(x_1,x_2)\to \alpha_0+\alpha_1x_1+\alpha_1x_2$$