Q.If demand and the cost function of a firm are $p=2-x$ and $c=-2x^2+2x+7$, then its profit function is :

Form revenue $R=px=2x-x^2$, then profit $P=R-c=x^2-7$.

The profit function is defined as

$P(x)=R(x)-c(x),\qquad \text{where } R(x)=p\cdot x.$

Step 1 — Revenue. With demand $p=2-x$,

$R=px=(2-x)x=2x-x^2.$

Step 2 — Profit. Subtract the cost $c=-2x^2+2x+7$:

$P=R-c=(2x-x^2)-(-2x^2+2x+7).$

$P=2x-x^2+2x^2-2x-7=x^2-7.$

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