Demand function from its price elasticity

Setting up. Price is written p, quantity x and the price elasticity η (Eₚ). The price elasticity of demand is η = −(p/x)(dx/dp), the positive number for a downward-sloping demand curve. Given η = p/[(p − 2)(p − 3)]:

−(p/x)(dx/dp) = p/[(p − 2)(p − 3)] ⟹ dx/x = −dp/[(p − 2)(p − 3)].

Integrating (deriving the demand function from the elasticity of demand, a Unit 3 integration topic). By partial fractions, 1/[(p − 2)(p − 3)] = 1/(p − 3) − 1/(p − 2). Hence, using ∫dx/x = log x,

log x = −log(p − 3) + log(p − 2) + log k ⟹ x = k(p − 2)/(p − 3) (for p > 3).

Constant from the given point. x = 6 when p = 4: 6 = k(2)/(1) ⟹ k = 3.

Demand function: x = 3(p − 2)/(p − 3), p > 3.

Check. dx/dp = 3[(p − 3) − (p − 2)]/(p − 3)² = −3/(p − 3)² < 0, so demand falls as price rises, and

−(p/x)(dx/dp) = [p(p − 3)/(3(p − 2))] × [3/(p − 3)²] = p/[(p − 2)(p − 3)], the given elasticity. At p = 4: x = 3 × 2/1 = 6.

The other reading. If the elasticity were defined without the minus sign, (p/x)(dx/dp) = p/[(p − 2)(p − 3)], the same steps give x = 12(p − 3)/(p − 2), which rises with price; it is not a demand law, so the signed (negative-slope) reading above is taken.