Functions (Single & Multivariable) and Economic Examples

1) Function: definition

A function $f$ from set $A$ to set $B$ assigns each $x\in A$ to exactly one $y\in B$ :

f: A \to B,\quad y=f(x)

Different inputs give different outputs:

x_1\ne x_2 \Rightarrow f(x_1)\ne f(x_2)

Every element of codomain is hit:

\forall y\in B,\exists x\in A \text{ s.t. } f(x)=y

Both one-one and onto → invertible.

y=a+bx

Used for demand/supply approximations, linear regressions.

y=ax^2+bx+c

Used for cost/revenue with turning points.

Y = A K^\alpha L^\beta

Special case with constant returns: $\alpha+\beta=1$ .

y = Ae^{rx}

Used in continuous growth, compounding.